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Pi approximation games

Discussion in 'Embedded' started by Tim Wescott, May 2, 2012.

  1. Tim Wescott

    Tim Wescott Guest

    Instead of doing productive work, I just spent a few enjoyable minutes
    with Scilab finding approximations to pi of the form m/n.

    Because I'm posting to a couple of nerd groups, I can be confident that
    most of you probably know 22/7 off the tops of your heads.

    What interested me is how spotty things are -- after 22/7, the error
    drops for a bit until you get down to 355/113 (which, if you're at an
    equal level of nerdiness to me will ring a bell, but not have been
    swimming around in your brain to be found).

    But what's _really_ interesting, is that the next better fit isn't found
    until you get up to 52163/16604. Then things get steadily better until
    you hit 104348/33215 -- at which point the next lowest ratio which
    improves anything is 208341/66317, then 312689/99532. At this point I
    decided that I would post my answers for your amusement, and get back to
    being productive.

    Discrete math is so fun. And these newfangled chips are just destroying
    the joy, by making floating point efficient and cheap enough that you
    don't need to know little tricks like pi = (almost) 355/113.

    --
    My liberal friends think I'm a conservative kook.
    My conservative friends think I'm a liberal kook.
    Why am I not happy that they have found common ground?

    Tim Wescott, Communications, Control, Circuits & Software
    http://www.wescottdesign.com
    Tim Wescott, May 2, 2012
    #1
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  2. Tim Wescott

    John S Guest

    On 5/1/2012 6:16 PM, Tim Wescott wrote:
    > Instead of doing productive work, I just spent a few enjoyable minutes
    > with Scilab finding approximations to pi of the form m/n.
    >
    > Because I'm posting to a couple of nerd groups, I can be confident that
    > most of you probably know 22/7 off the tops of your heads.
    >
    > What interested me is how spotty things are -- after 22/7, the error
    > drops for a bit until you get down to 355/113 (which, if you're at an
    > equal level of nerdiness to me will ring a bell, but not have been
    > swimming around in your brain to be found).
    >
    > But what's _really_ interesting, is that the next better fit isn't found
    > until you get up to 52163/16604. Then things get steadily better until
    > you hit 104348/33215 -- at which point the next lowest ratio which
    > improves anything is 208341/66317, then 312689/99532. At this point I
    > decided that I would post my answers for your amusement, and get back to
    > being productive.
    >
    > Discrete math is so fun. And these newfangled chips are just destroying
    > the joy, by making floating point efficient and cheap enough that you
    > don't need to know little tricks like pi = (almost) 355/113.
    >


    I like the idea that both 22 and 7 each fit into a byte whereas 355 does
    not. And, 22/7 is hi by only .04%. Beautiful!

    John S
    John S, May 2, 2012
    #2
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  3. Tim Wescott

    Tim Wescott Guest

    On Tue, 01 May 2012 18:21:29 -0500, John S wrote:

    > On 5/1/2012 6:16 PM, Tim Wescott wrote:
    >> Instead of doing productive work, I just spent a few enjoyable minutes
    >> with Scilab finding approximations to pi of the form m/n.
    >>
    >> Because I'm posting to a couple of nerd groups, I can be confident that
    >> most of you probably know 22/7 off the tops of your heads.
    >>
    >> What interested me is how spotty things are -- after 22/7, the error
    >> drops for a bit until you get down to 355/113 (which, if you're at an
    >> equal level of nerdiness to me will ring a bell, but not have been
    >> swimming around in your brain to be found).
    >>
    >> But what's _really_ interesting, is that the next better fit isn't
    >> found until you get up to 52163/16604. Then things get steadily better
    >> until you hit 104348/33215 -- at which point the next lowest ratio
    >> which improves anything is 208341/66317, then 312689/99532. At this
    >> point I decided that I would post my answers for your amusement, and
    >> get back to being productive.
    >>
    >> Discrete math is so fun. And these newfangled chips are just
    >> destroying the joy, by making floating point efficient and cheap enough
    >> that you don't need to know little tricks like pi = (almost) 355/113.
    >>
    >>

    > I like the idea that both 22 and 7 each fit into a byte whereas 355 does
    > not. And, 22/7 is hi by only .04%. Beautiful!
    >
    > John S


    245/78. It's only a bit better than twice as good as 22/7 -- then along
    comes 355/113, which is over 1000 times better than 245/78.

    --
    My liberal friends think I'm a conservative kook.
    My conservative friends think I'm a liberal kook.
    Why am I not happy that they have found common ground?

    Tim Wescott, Communications, Control, Circuits & Software
    http://www.wescottdesign.com
    Tim Wescott, May 2, 2012
    #3
  4. Tim Wescott

    Joel Koltner Guest

    Tim Wescott wrote:
    > Discrete math is so fun. And these newfangled chips are just destroying
    > the joy, by making floating point efficient and cheap enough that you
    > don't need to know little tricks like pi = (almost) 355/113.


    --> http://xkcd.com/1047/

    :)
    Joel Koltner, May 2, 2012
    #4
  5. Tim Wescott

    Joel Koltner Guest

    John S wrote:
    > I like the idea that both 22 and 7 each fit into a byte whereas 355 does
    > not. And, 22/7 is hi by only .04%. Beautiful!


    Jack Crenshaw's book, "Math Toolkit for Real-Time Programming"
    (http://www.amazon.com/Math-Toolkit-Real-Time-Programming-ebook/dp/B003WUYQVY)
    spends a lot of time discussing how to make "good enough" approximations
    of various, e.g., transcendental functions... and how to know when "good
    enough" really is. It's quite handy for this sort of thing...
    Joel Koltner, May 2, 2012
    #5
  6. Tim Wescott

    Steve Pope Guest

    Tim Wescott <> wrote:

    >On Tue, 01 May 2012 18:21:29 -0500, John S wrote:


    >> On 5/1/2012 6:16 PM, Tim Wescott wrote:
    >>> Instead of doing productive work, I just spent a few enjoyable minutes
    >>> with Scilab finding approximations to pi of the form m/n.
    >>>
    >>> Because I'm posting to a couple of nerd groups, I can be confident that
    >>> most of you probably know 22/7 off the tops of your heads.
    >>>
    >>> What interested me is how spotty things are -- after 22/7, the error
    >>> drops for a bit until you get down to 355/113 (which, if you're at an
    >>> equal level of nerdiness to me will ring a bell, but not have been
    >>> swimming around in your brain to be found).
    >>>
    >>> But what's _really_ interesting, is that the next better fit isn't
    >>> found until you get up to 52163/16604. Then things get steadily better
    >>> until you hit 104348/33215 -- at which point the next lowest ratio
    >>> which improves anything is 208341/66317, then 312689/99532. At this
    >>> point I decided that I would post my answers for your amusement, and
    >>> get back to being productive.
    >>>
    >>> Discrete math is so fun. And these newfangled chips are just
    >>> destroying the joy, by making floating point efficient and cheap enough
    >>> that you don't need to know little tricks like pi = (almost) 355/113.
    >>>
    >>>

    >> I like the idea that both 22 and 7 each fit into a byte whereas 355 does
    >> not. And, 22/7 is hi by only .04%. Beautiful!
    >>
    >> John S

    >
    >245/78. It's only a bit better than twice as good as 22/7 -- then along
    >comes 355/113, which is over 1000 times better than 245/78.


    Suppose you do the same thing with the fine structure constant --
    let me know what you discover.


    Steve
    Steve Pope, May 2, 2012
    #6
  7. Tim Wescott

    John Larkin Guest

    On Tue, 01 May 2012 18:16:25 -0500, Tim Wescott <>
    wrote:

    >Instead of doing productive work, I just spent a few enjoyable minutes
    >with Scilab finding approximations to pi of the form m/n.
    >
    >Because I'm posting to a couple of nerd groups, I can be confident that
    >most of you probably know 22/7 off the tops of your heads.
    >
    >What interested me is how spotty things are -- after 22/7, the error
    >drops for a bit until you get down to 355/113 (which, if you're at an
    >equal level of nerdiness to me will ring a bell, but not have been
    >swimming around in your brain to be found).
    >
    >But what's _really_ interesting, is that the next better fit isn't found
    >until you get up to 52163/16604. Then things get steadily better until
    >you hit 104348/33215 -- at which point the next lowest ratio which
    >improves anything is 208341/66317, then 312689/99532. At this point I
    >decided that I would post my answers for your amusement, and get back to
    >being productive.
    >
    >Discrete math is so fun. And these newfangled chips are just destroying
    >the joy, by making floating point efficient and cheap enough that you
    >don't need to know little tricks like pi = (almost) 355/113.


    I once knew pi to 100 places, but now I've forgotten everything past
    19.


    --

    John Larkin Highland Technology, Inc

    jlarkin at highlandtechnology dot com
    http://www.highlandtechnology.com

    Precision electronic instrumentation
    Picosecond-resolution Digital Delay and Pulse generators
    Custom laser drivers and controllers
    Photonics and fiberoptic TTL data links
    VME thermocouple, LVDT, synchro acquisition and simulation
    John Larkin, May 2, 2012
    #7
  8. Tim Wescott

    John Larkin Guest

    On Tue, 01 May 2012 18:16:25 -0500, Tim Wescott <>
    wrote:

    >Instead of doing productive work, I just spent a few enjoyable minutes
    >with Scilab finding approximations to pi of the form m/n.
    >
    >Because I'm posting to a couple of nerd groups, I can be confident that
    >most of you probably know 22/7 off the tops of your heads.
    >
    >What interested me is how spotty things are -- after 22/7, the error
    >drops for a bit until you get down to 355/113 (which, if you're at an
    >equal level of nerdiness to me will ring a bell, but not have been
    >swimming around in your brain to be found).
    >
    >But what's _really_ interesting, is that the next better fit isn't found
    >until you get up to 52163/16604. Then things get steadily better until
    >you hit 104348/33215 -- at which point the next lowest ratio which
    >improves anything is 208341/66317, then 312689/99532. At this point I
    >decided that I would post my answers for your amusement, and get back to
    >being productive.
    >
    >Discrete math is so fun. And these newfangled chips are just destroying
    >the joy, by making floating point efficient and cheap enough that you
    >don't need to know little tricks like pi = (almost) 355/113.


    My old HP35 calculators have a key for pi. The newer ones hide it, a
    tiny pastel shift key thing. So I just key in 3.14. Rob down the hall
    uses 3.

    We are increasingly using floats in embedded stuff. Our ARM LPC3250
    has SIMD hardware FP operations.


    --

    John Larkin Highland Technology, Inc

    jlarkin at highlandtechnology dot com
    http://www.highlandtechnology.com

    Precision electronic instrumentation
    Picosecond-resolution Digital Delay and Pulse generators
    Custom laser drivers and controllers
    Photonics and fiberoptic TTL data links
    VME thermocouple, LVDT, synchro acquisition and simulation
    John Larkin, May 2, 2012
    #8
  9. Tim Wescott

    John S Guest

    On 5/1/2012 6:28 PM, Tim Wescott wrote:
    > On Tue, 01 May 2012 18:21:29 -0500, John S wrote:
    >
    >> On 5/1/2012 6:16 PM, Tim Wescott wrote:
    >>> Instead of doing productive work, I just spent a few enjoyable minutes
    >>> with Scilab finding approximations to pi of the form m/n.
    >>>
    >>> Because I'm posting to a couple of nerd groups, I can be confident that
    >>> most of you probably know 22/7 off the tops of your heads.
    >>>
    >>> What interested me is how spotty things are -- after 22/7, the error
    >>> drops for a bit until you get down to 355/113 (which, if you're at an
    >>> equal level of nerdiness to me will ring a bell, but not have been
    >>> swimming around in your brain to be found).
    >>>
    >>> But what's _really_ interesting, is that the next better fit isn't
    >>> found until you get up to 52163/16604. Then things get steadily better
    >>> until you hit 104348/33215 -- at which point the next lowest ratio
    >>> which improves anything is 208341/66317, then 312689/99532. At this
    >>> point I decided that I would post my answers for your amusement, and
    >>> get back to being productive.
    >>>
    >>> Discrete math is so fun. And these newfangled chips are just
    >>> destroying the joy, by making floating point efficient and cheap enough
    >>> that you don't need to know little tricks like pi = (almost) 355/113.
    >>>
    >>>

    >> I like the idea that both 22 and 7 each fit into a byte whereas 355 does
    >> not. And, 22/7 is hi by only .04%. Beautiful!
    >>
    >> John S

    >
    > 245/78. It's only a bit better than twice as good as 22/7 -- then along
    > comes 355/113, which is over 1000 times better than 245/78.
    >


    245/78 is more easily forgotten.
    John S, May 2, 2012
    #9
  10. Tim Wescott

    Les Cargill Guest

    John S wrote:
    > On 5/1/2012 6:28 PM, Tim Wescott wrote:
    >> On Tue, 01 May 2012 18:21:29 -0500, John S wrote:
    >>
    >>> On 5/1/2012 6:16 PM, Tim Wescott wrote:
    >>>> Instead of doing productive work, I just spent a few enjoyable minutes
    >>>> with Scilab finding approximations to pi of the form m/n.
    >>>>
    >>>> Because I'm posting to a couple of nerd groups, I can be confident that
    >>>> most of you probably know 22/7 off the tops of your heads.
    >>>>
    >>>> What interested me is how spotty things are -- after 22/7, the error
    >>>> drops for a bit until you get down to 355/113 (which, if you're at an
    >>>> equal level of nerdiness to me will ring a bell, but not have been
    >>>> swimming around in your brain to be found).
    >>>>
    >>>> But what's _really_ interesting, is that the next better fit isn't
    >>>> found until you get up to 52163/16604. Then things get steadily better
    >>>> until you hit 104348/33215 -- at which point the next lowest ratio
    >>>> which improves anything is 208341/66317, then 312689/99532. At this
    >>>> point I decided that I would post my answers for your amusement, and
    >>>> get back to being productive.
    >>>>
    >>>> Discrete math is so fun. And these newfangled chips are just
    >>>> destroying the joy, by making floating point efficient and cheap enough
    >>>> that you don't need to know little tricks like pi = (almost) 355/113.
    >>>>
    >>>>
    >>> I like the idea that both 22 and 7 each fit into a byte whereas 355 does
    >>> not. And, 22/7 is hi by only .04%. Beautiful!
    >>>
    >>> John S

    >>
    >> 245/78. It's only a bit better than twice as good as 22/7 -- then along
    >> comes 355/113, which is over 1000 times better than 245/78.
    >>

    >
    > 245/78 is more easily forgotten.



    but highly mnenomic - it's 2345678 with the 3 dropped and the 6 turned
    into a divide sign...

    --
    Les Cargill
    Les Cargill, May 2, 2012
    #10
  11. Tim Wescott

    Nico Coesel Guest

    Tim Wescott <> wrote:

    >Instead of doing productive work, I just spent a few enjoyable minutes
    >with Scilab finding approximations to pi of the form m/n.
    >
    >Because I'm posting to a couple of nerd groups, I can be confident that
    >most of you probably know 22/7 off the tops of your heads.


    Now you mention it :)

    >Discrete math is so fun. And these newfangled chips are just destroying
    >the joy, by making floating point efficient and cheap enough that you
    >don't need to know little tricks like pi = (almost) 355/113.


    You can always declare a constant or use pi=4*arctan(1) although I
    seldomly see the latter.

    --
    Failure does not prove something is impossible, failure simply
    indicates you are not using the right tools...
    nico@nctdevpuntnl (punt=.)
    --------------------------------------------------------------
    Nico Coesel, May 2, 2012
    #11
  12. Tim Wescott

    John S Guest

    On 5/1/2012 6:30 PM, Joel Koltner wrote:
    > Tim Wescott wrote:
    >> Discrete math is so fun. And these newfangled chips are just destroying
    >> the joy, by making floating point efficient and cheap enough that you
    >> don't need to know little tricks like pi = (almost) 355/113.

    >
    > --> http://xkcd.com/1047/
    >
    > :)
    >


    Off the subject, but this one is really funny...

    http://xkcd.com/1020/
    John S, May 2, 2012
    #12
  13. Tim Wescott

    Randy Yates Guest

    John S <> writes:

    > On 5/1/2012 6:30 PM, Joel Koltner wrote:
    >> Tim Wescott wrote:
    >>> Discrete math is so fun. And these newfangled chips are just destroying
    >>> the joy, by making floating point efficient and cheap enough that you
    >>> don't need to know little tricks like pi = (almost) 355/113.

    >>
    >> --> http://xkcd.com/1047/
    >>
    >> :)
    >>

    >
    > Off the subject, but this one is really funny...
    >
    > http://xkcd.com/1020/


    That's mean - at the end "not all these are wrong" - several looked
    close (at least in my head).
    --
    Randy Yates
    DSP/Firmware Engineer
    919-577-9882 (H)
    919-720-2916 (C)
    Randy Yates, May 2, 2012
    #13
  14. In comp.dsp Tim Wescott <> wrote:
    > Instead of doing productive work, I just spent a few enjoyable minutes
    > with Scilab finding approximations to pi of the form m/n.


    > Because I'm posting to a couple of nerd groups, I can be confident
    > that most of you probably know 22/7 off the tops of your heads.


    > What interested me is how spotty things are -- after 22/7, the error
    > drops for a bit until you get down to 355/113 (which, if you're at an
    > equal level of nerdiness to me will ring a bell, but not have been
    > swimming around in your brain to be found).


    Yes. It was the first problem in the book for the HP 25C calculator
    that I got many years ago.

    > But what's _really_ interesting, is that the next better fit
    > isn't found until you get up to 52163/16604. Then things get
    > steadily better until you hit 104348/33215 -- at which point
    > the next lowest ratio which improves anything is 208341/66317,
    > then 312689/99532. At this point I decided that I would post
    > my answers for your amusement, and get back to being productive.


    There is an algorithm that some calculators use for converting a
    decimal result to a fraction. If I remember, that one easily finds
    successively better fractions approximating any given value.

    I don't remember the details, but I do remember how funny it is,
    in that at one point it takes two fractions, and adds their numerators
    and denominators, before goint to the next step.

    I believe it is described in the manual for one of the HP
    calculators that does that conversion.

    Otherwise, I have the TI-92, which will generate fraction (rational)
    results, then you ask for an approximate result. Some calculations
    will give a symbolic pi result.

    > Discrete math is so fun. And these newfangled chips are just
    > destroying the joy, by making floating point efficient and
    > cheap enough that you don't need to know little tricks
    > like pi = (almost) 355/113.


    -- glen
    glen herrmannsfeldt, May 2, 2012
    #14
  15. On Tue, 01 May 2012 18:16:25 -0500, Tim Wescott <>
    wrote:

    >Instead of doing productive work, I just spent a few enjoyable minutes
    >with Scilab finding approximations to pi of the form m/n.
    >
    >Because I'm posting to a couple of nerd groups, I can be confident that
    >most of you probably know 22/7 off the tops of your heads.
    >
    >What interested me is how spotty things are -- after 22/7, the error
    >drops for a bit until you get down to 355/113 (which, if you're at an
    >equal level of nerdiness to me will ring a bell, but not have been
    >swimming around in your brain to be found).
    >
    >But what's _really_ interesting, is that the next better fit isn't found
    >until you get up to 52163/16604. Then things get steadily better until
    >you hit 104348/33215 -- at which point the next lowest ratio which
    >improves anything is 208341/66317, then 312689/99532. At this point I
    >decided that I would post my answers for your amusement, and get back to
    >being productive.
    >
    >Discrete math is so fun. And these newfangled chips are just destroying
    >the joy, by making floating point efficient and cheap enough that you
    >don't need to know little tricks like pi = (almost) 355/113.


    Hi Tim,

    You may find the material linked to from this page of interest:

    http://www.dtashley.com/howtos/2007/01/best_rational_approximation/

    Generally, the rational numbers you mentioned are convergents to PI. I
    believe that Chapter 2 of Khinchin (linked to from the URL above),
    mentions convergents as best approximations.

    Any rational number has a straightforward way to find the continued
    fraction representation (and hence the convergents), and it is a
    simple technique you can do on the back of a napkin.

    However, irrational numbers require symbolic techniques. You can do
    sqrt(2) easily, but I'm not sure that you can symbolically tackle PI
    in the same sense.

    Anyway, using the program linked to above and a bunch of digits of PI
    from another website, I get the output below.

    The convergents are 3/1, 22/7, 333/106, 355/113, etc. You left a lot
    of convergents out of your discussion.

    In any case, the number theorists have done this all before ... you
    are not the first.

    DTA
    ----------

    ------------------------------------------------------------------------------
    MAJOR MODE: Finding closest rational number(s) under the constraints.
    ------------------------------------------------------------------------------
    RI_IN Numerator: 1,570,796,326,794,896,619,231, ( 76
    digits)
    321,691,639,751,442,098,584,699,687,
    552,910,487,472,296,153,908,203,143
    ------------------------------------------------------------------------------
    RI_IN Denominator: 500,000,000,000,000,000,000, ( 75
    digits)
    000,000,000,000,000,000,000,000,000,
    000,000,000,000,000,000,000,000,000
    ------------------------------------------------------------------------------
    K_MAX: 99,999,999,999,999,999,999, ( 47
    digits)
    999,999,999,999,999,999,999,999,999
    ------------------------------------------------------------------------------
    H_MAX: 9,999,999,999,999,999,999, ( 46
    digits)
    999,999,999,999,999,999,999,999,999
    ------------------------------------------------------------------------------
    ******************************************************************************
    ********* CF Rep Of Reciprocal Of Abs Value Of Number Specified
    *********
    ******************************************************************************
    Input Numerator: 500,000,000,000,000,000,000, ( 75
    digits)
    000,000,000,000,000,000,000,000,000,
    000,000,000,000,000,000,000,000,000
    ------------------------------------------------------------------------------
    Input Denominator: 1,570,796,326,794,896,619,231, ( 76
    digits)
    321,691,639,751,442,098,584,699,687,
    552,910,487,472,296,153,908,203,143
    ------------------------------------------------------------------------------
    dividend(0): 500,000,000,000,000,000,000, ( 75
    digits)
    000,000,000,000,000,000,000,000,000,
    000,000,000,000,000,000,000,000,000
    ------------------------------------------------------------------------------
    divisor(0): 1,570,796,326,794,896,619,231, ( 76
    digits)
    321,691,639,751,442,098,584,699,687,
    552,910,487,472,296,153,908,203,143
    ------------------------------------------------------------------------------
    a(0): 0 ( 1
    digit )
    ------------------------------------------------------------------------------
    p(0): 0 ( 1
    digit )
    ------------------------------------------------------------------------------
    q(0): 1 ( 1
    digit )
    ------------------------------------------------------------------------------
    dap_h(0): 0 ( 1
    digit )
    ------------------------------------------------------------------------------
    dap_k(0): 1, ( 109
    digits)
    000,000,000,000,000,000,000,000,000,
    000,000,000,000,000,000,000,000,000,
    000,000,000,000,000,000,000,000,000,
    000,000,000,000,000,000,000,000,000
    ------------------------------------------------------------------------------
    dividend(1): 1,570,796,326,794,896,619,231, ( 76
    digits)
    321,691,639,751,442,098,584,699,687,
    552,910,487,472,296,153,908,203,143
    ------------------------------------------------------------------------------
    divisor(1): 500,000,000,000,000,000,000, ( 75
    digits)
    000,000,000,000,000,000,000,000,000,
    000,000,000,000,000,000,000,000,000
    ------------------------------------------------------------------------------
    a(1): 3 ( 1
    digit )
    ------------------------------------------------------------------------------
    p(1): 1 ( 1
    digit )
    ------------------------------------------------------------------------------
    q(1): 3 ( 1
    digit )
    ------------------------------------------------------------------------------
    dap_h(1): 333,333,333,333,333,333,333,333,333, ( 108
    digits)
    333,333,333,333,333,333,333,333,333,
    333,333,333,333,333,333,333,333,333,
    333,333,333,333,333,333,333,333,333
    ------------------------------------------------------------------------------
    dap_k(1): 1, ( 109
    digits)
    000,000,000,000,000,000,000,000,000,
    000,000,000,000,000,000,000,000,000,
    000,000,000,000,000,000,000,000,000,
    000,000,000,000,000,000,000,000,000
    ------------------------------------------------------------------------------
    dividend(2): 500,000,000,000,000,000,000, ( 75
    digits)
    000,000,000,000,000,000,000,000,000,
    000,000,000,000,000,000,000,000,000
    ------------------------------------------------------------------------------
    divisor(2): 70,796,326,794,896,619,231, ( 74
    digits)
    321,691,639,751,442,098,584,699,687,
    552,910,487,472,296,153,908,203,143
    ------------------------------------------------------------------------------
    a(2): 7 ( 1
    digit )
    ------------------------------------------------------------------------------
    p(2): 7 ( 1
    digit )
    ------------------------------------------------------------------------------
    q(2): 22 ( 2
    digits)
    ------------------------------------------------------------------------------
    dap_h(2): 318,181,818,181,818,181,818,181,818, ( 108
    digits)
    181,818,181,818,181,818,181,818,181,
    818,181,818,181,818,181,818,181,818,
    181,818,181,818,181,818,181,818,181
    ------------------------------------------------------------------------------
    dap_k(2): 1, ( 109
    digits)
    000,000,000,000,000,000,000,000,000,
    000,000,000,000,000,000,000,000,000,
    000,000,000,000,000,000,000,000,000,
    000,000,000,000,000,000,000,000,000
    ------------------------------------------------------------------------------
    dividend(3): 70,796,326,794,896,619,231, ( 74
    digits)
    321,691,639,751,442,098,584,699,687,
    552,910,487,472,296,153,908,203,143
    ------------------------------------------------------------------------------
    divisor(3): 4,425,712,435,723,665,380, ( 73
    digits)
    748,158,521,739,905,309,907,102,187,
    129,626,587,693,926,922,642,577,999
    ------------------------------------------------------------------------------
    a(3): 15 ( 2
    digits)
    ------------------------------------------------------------------------------
    p(3): 106 ( 3
    digits)
    ------------------------------------------------------------------------------
    q(3): 333 ( 3
    digits)
    ------------------------------------------------------------------------------
    dap_h(3): 318,318,318,318,318,318,318,318,318, ( 108
    digits)
    318,318,318,318,318,318,318,318,318,
    318,318,318,318,318,318,318,318,318,
    318,318,318,318,318,318,318,318,318
    ------------------------------------------------------------------------------
    dap_k(3): 1, ( 109
    digits)
    000,000,000,000,000,000,000,000,000,
    000,000,000,000,000,000,000,000,000,
    000,000,000,000,000,000,000,000,000,
    000,000,000,000,000,000,000,000,000
    ------------------------------------------------------------------------------
    dividend(4): 4,425,712,435,723,665,380, ( 73
    digits)
    748,158,521,739,905,309,907,102,187,
    129,626,587,693,926,922,642,577,999
    ------------------------------------------------------------------------------
    divisor(4): 4,410,640,259,041,638,520, ( 73
    digits)
    099,313,813,652,862,449,978,166,880,
    608,511,672,063,392,314,269,533,158
    ------------------------------------------------------------------------------
    a(4): 1 ( 1
    digit )
    ------------------------------------------------------------------------------
    p(4): 113 ( 3
    digits)
    ------------------------------------------------------------------------------
    q(4): 355 ( 3
    digits)
    ------------------------------------------------------------------------------
    dap_h(4): 318,309,859,154,929,577,464,788,732, ( 108
    digits)
    394,366,197,183,098,591,549,295,774,
    647,887,323,943,661,971,830,985,915,
    492,957,746,478,873,239,436,619,718
    ------------------------------------------------------------------------------
    dap_k(4): 1, ( 109
    digits)
    000,000,000,000,000,000,000,000,000,
    000,000,000,000,000,000,000,000,000,
    000,000,000,000,000,000,000,000,000,
    000,000,000,000,000,000,000,000,000
    ------------------------------------------------------------------------------
    dividend(5): 4,410,640,259,041,638,520, ( 73
    digits)
    099,313,813,652,862,449,978,166,880,
    608,511,672,063,392,314,269,533,158
    ------------------------------------------------------------------------------
    divisor(5): 15,072,176,682,026,860, ( 71
    digits)
    648,844,708,087,042,859,928,935,306,
    521,114,915,630,534,608,373,044,841
    ------------------------------------------------------------------------------
    a(5): 292 ( 3
    digits)
    ------------------------------------------------------------------------------
    p(5): 33,102 ( 5
    digits)
    ------------------------------------------------------------------------------
    q(5): 103,993 ( 6
    digits)
    ------------------------------------------------------------------------------
    dap_h(5): 318,309,886,242,343,234,640,793,130, ( 108
    digits)
    306,847,576,279,172,636,619,772,484,
    686,469,281,586,260,613,695,152,558,
    345,273,239,544,969,372,938,563,172
    ------------------------------------------------------------------------------
    dap_k(5): 1, ( 109
    digits)
    000,000,000,000,000,000,000,000,000,
    000,000,000,000,000,000,000,000,000,
    000,000,000,000,000,000,000,000,000,
    000,000,000,000,000,000,000,000,000
    ------------------------------------------------------------------------------
    dividend(6): 15,072,176,682,026,860, ( 71
    digits)
    648,844,708,087,042,859,928,935,306,
    521,114,915,630,534,608,373,044,841
    ------------------------------------------------------------------------------
    divisor(6): 9,564,667,889,795,210, ( 70
    digits)
    636,659,052,236,347,350,729,057,376,
    442,956,307,947,286,669,340,439,586
    ------------------------------------------------------------------------------
    a(6): 1 ( 1
    digit )
    ------------------------------------------------------------------------------
    p(6): 33,215 ( 5
    digits)
    ------------------------------------------------------------------------------
    q(6): 104,348 ( 6
    digits)
    ------------------------------------------------------------------------------
    dap_h(6): 318,309,886,150,189,749,683,750,527, ( 108
    digits)
    082,454,862,575,229,041,284,931,191,
    781,347,031,088,281,519,530,800,781,
    998,696,668,838,885,268,524,552,459
    ------------------------------------------------------------------------------
    dap_k(6): 1, ( 109
    digits)
    000,000,000,000,000,000,000,000,000,
    000,000,000,000,000,000,000,000,000,
    000,000,000,000,000,000,000,000,000,
    000,000,000,000,000,000,000,000,000
    ------------------------------------------------------------------------------
    dividend(7): 9,564,667,889,795,210, ( 70
    digits)
    636,659,052,236,347,350,729,057,376,
    442,956,307,947,286,669,340,439,586
    ------------------------------------------------------------------------------
    divisor(7): 5,507,508,792,231,650, ( 70
    digits)
    012,185,655,850,695,509,199,877,930,
    078,158,607,683,247,939,032,605,255
    ------------------------------------------------------------------------------
    a(7): 1 ( 1
    digit )
    ------------------------------------------------------------------------------
    p(7): 66,317 ( 5
    digits)
    ------------------------------------------------------------------------------
    q(7): 208,341 ( 6
    digits)
    ------------------------------------------------------------------------------
    dap_h(7): 318,309,886,196,187,980,282,325,610, ( 108
    digits)
    417,536,634,651,844,812,110,914,318,
    353,084,606,486,481,297,488,252,432,
    310,490,973,932,159,296,537,887,405
    ------------------------------------------------------------------------------
    dap_k(7): 1, ( 109
    digits)
    000,000,000,000,000,000,000,000,000,
    000,000,000,000,000,000,000,000,000,
    000,000,000,000,000,000,000,000,000,
    000,000,000,000,000,000,000,000,000
    ------------------------------------------------------------------------------
    dividend(8): 5,507,508,792,231,650, ( 70
    digits)
    012,185,655,850,695,509,199,877,930,
    078,158,607,683,247,939,032,605,255
    ------------------------------------------------------------------------------
    divisor(8): 4,057,159,097,563,560, ( 70
    digits)
    624,473,396,385,651,841,529,179,446,
    364,797,700,264,038,730,307,834,331
    ------------------------------------------------------------------------------
    a(8): 1 ( 1
    digit )
    ------------------------------------------------------------------------------
    p(8): 99,532 ( 5
    digits)
    ------------------------------------------------------------------------------
    q(8): 312,689 ( 6
    digits)
    ------------------------------------------------------------------------------
    dap_h(8): 318,309,886,180,837,829,280,850,941, ( 108
    digits)
    350,671,114,110,186,159,410,788,355,
    202,773,362,670,257,028,549,133,484,
    068,835,168,490,097,189,219,959,768
    ------------------------------------------------------------------------------
    dap_k(8): 1, ( 109
    digits)
    000,000,000,000,000,000,000,000,000,
    000,000,000,000,000,000,000,000,000,
    000,000,000,000,000,000,000,000,000,
    000,000,000,000,000,000,000,000,000
    ------------------------------------------------------------------------------
    dividend(9): 4,057,159,097,563,560, ( 70
    digits)
    624,473,396,385,651,841,529,179,446,
    364,797,700,264,038,730,307,834,331
    ------------------------------------------------------------------------------
    divisor(9): 1,450,349,694,668,089, ( 70
    digits)
    387,712,259,465,043,667,670,698,483,
    713,360,907,419,209,208,724,770,924
    ------------------------------------------------------------------------------
    a(9): 2 ( 1
    digit )
    ------------------------------------------------------------------------------
    p(9): 265,381 ( 6
    digits)
    ------------------------------------------------------------------------------
    q(9): 833,719 ( 6
    digits)
    ------------------------------------------------------------------------------
    dap_h(9): 318,309,886,184,673,732,996,369,280, ( 108
    digits)
    297,078,512,064,616,495,485,889,130,
    510,399,786,978,586,310,255,613,701,
    978,724,246,418,757,399,075,707,762
    ------------------------------------------------------------------------------
    dap_k(9): 1, ( 109
    digits)
    000,000,000,000,000,000,000,000,000,
    000,000,000,000,000,000,000,000,000,
    000,000,000,000,000,000,000,000,000,
    000,000,000,000,000,000,000,000,000
    ------------------------------------------------------------------------------
    dividend(10): 1,450,349,694,668,089, ( 70
    digits)
    387,712,259,465,043,667,670,698,483,
    713,360,907,419,209,208,724,770,924
    ------------------------------------------------------------------------------
    divisor(10): 1,156,459,708,227,381, ( 70
    digits)
    849,048,877,455,564,506,187,782,478,
    938,075,885,425,620,312,858,292,483
    ------------------------------------------------------------------------------
    a(10): 1 ( 1
    digit )
    ------------------------------------------------------------------------------
    p(10): 364,913 ( 6
    digits)
    ------------------------------------------------------------------------------
    q(10): 1,146,408 ( 7
    digits)
    ------------------------------------------------------------------------------
    dap_h(10): 318,309,886,183,627,469,452,411,357, ( 108
    digits)
    910,970,614,301,365,656,903,999,274,
    254,890,056,594,161,938,855,974,487,
    268,058,143,348,615,850,552,333,898
    ------------------------------------------------------------------------------
    dap_k(10): 1, ( 109
    digits)
    000,000,000,000,000,000,000,000,000,
    000,000,000,000,000,000,000,000,000,
    000,000,000,000,000,000,000,000,000,
    000,000,000,000,000,000,000,000,000
    ------------------------------------------------------------------------------
    dividend(11): 1,156,459,708,227,381, ( 70
    digits)
    849,048,877,455,564,506,187,782,478,
    938,075,885,425,620,312,858,292,483
    ------------------------------------------------------------------------------
    divisor(11): 293,889,986,440,707, ( 69
    digits)
    538,663,382,009,479,161,482,916,004,
    775,285,021,993,588,895,866,478,441
    ------------------------------------------------------------------------------
    a(11): 3 ( 1
    digit )
    ------------------------------------------------------------------------------
    p(11): 1,360,120 ( 7
    digits)
    ------------------------------------------------------------------------------
    q(11): 4,272,943 ( 7
    digits)
    ------------------------------------------------------------------------------
    dap_h(11): 318,309,886,183,831,612,076,266,872, ( 108
    digits)
    738,531,733,280,785,631,823,312,410,
    205,331,547,834,829,530,840,921,584,
    959,125,361,606,742,706,373,569,691
    ------------------------------------------------------------------------------
    dap_k(11): 1, ( 109
    digits)
    000,000,000,000,000,000,000,000,000,
    000,000,000,000,000,000,000,000,000,
    000,000,000,000,000,000,000,000,000,
    000,000,000,000,000,000,000,000,000
    ------------------------------------------------------------------------------
    dividend(12): 293,889,986,440,707, ( 69
    digits)
    538,663,382,009,479,161,482,916,004,
    775,285,021,993,588,895,866,478,441
    ------------------------------------------------------------------------------
    divisor(12): 274,789,748,905,259, ( 69
    digits)
    233,058,731,427,127,021,739,034,464,
    612,220,819,444,853,625,258,857,160
    ------------------------------------------------------------------------------
    a(12): 1 ( 1
    digit )
    ------------------------------------------------------------------------------
    p(12): 1,725,033 ( 7
    digits)
    ------------------------------------------------------------------------------
    q(12): 5,419,351 ( 7
    digits)
    ------------------------------------------------------------------------------
    dap_h(12): 318,309,886,183,788,427,802,517,312, ( 108
    digits)
    497,382,066,598,011,459,305,736,055,
    848,753,845,248,259,431,802,811,812,
    705,986,381,026,067,512,512,107,077
    ------------------------------------------------------------------------------
    dap_k(12): 1, ( 109
    digits)
    000,000,000,000,000,000,000,000,000,
    000,000,000,000,000,000,000,000,000,
    000,000,000,000,000,000,000,000,000,
    000,000,000,000,000,000,000,000,000
    ------------------------------------------------------------------------------
    dividend(13): 274,789,748,905,259, ( 69
    digits)
    233,058,731,427,127,021,739,034,464,
    612,220,819,444,853,625,258,857,160
    ------------------------------------------------------------------------------
    divisor(13): 19,100,237,535,448, ( 68
    digits)
    305,604,650,582,352,139,743,881,540,
    163,064,202,548,735,270,607,621,281
    ------------------------------------------------------------------------------
    a(13): 14 ( 2
    digits)
    ------------------------------------------------------------------------------
    p(13): 25,510,582 ( 8
    digits)
    ------------------------------------------------------------------------------
    q(13): 80,143,857 ( 8
    digits)
    ------------------------------------------------------------------------------
    dap_h(13): 318,309,886,183,790,730,211,549,464, ( 108
    digits)
    109,270,408,585,401,623,483,132,337,
    890,850,449,086,821,464,058,062,491,
    302,358,956,844,315,591,150,049,092
    ------------------------------------------------------------------------------
    dap_k(13): 1, ( 109
    digits)
    000,000,000,000,000,000,000,000,000,
    000,000,000,000,000,000,000,000,000,
    000,000,000,000,000,000,000,000,000,
    000,000,000,000,000,000,000,000,000
    ------------------------------------------------------------------------------
    dividend(14): 19,100,237,535,448, ( 68
    digits)
    305,604,650,582,352,139,743,881,540,
    163,064,202,548,735,270,607,621,281
    ------------------------------------------------------------------------------
    divisor(14): 7,386,423,408,982, ( 67
    digits)
    954,593,623,274,197,065,324,692,902,
    329,321,983,762,559,836,752,159,226
    ------------------------------------------------------------------------------
    a(14): 2 ( 1
    digit )
    ------------------------------------------------------------------------------
    p(14): 52,746,197 ( 8
    digits)
    ------------------------------------------------------------------------------
    q(14): 165,707,065 ( 9
    digits)
    ------------------------------------------------------------------------------
    dap_h(14): 318,309,886,183,790,654,912,631,516, ( 108
    digits)
    344,822,111,235,872,773,439,080,584,
    765,652,568,887,150,345,701,916,813,
    263,212,404,371,533,585,487,136,592
    ------------------------------------------------------------------------------
    dap_k(14): 1, ( 109
    digits)
    000,000,000,000,000,000,000,000,000,
    000,000,000,000,000,000,000,000,000,
    000,000,000,000,000,000,000,000,000,
    000,000,000,000,000,000,000,000,000
    ------------------------------------------------------------------------------
    dividend(15): 7,386,423,408,982, ( 67
    digits)
    954,593,623,274,197,065,324,692,902,
    329,321,983,762,559,836,752,159,226
    ------------------------------------------------------------------------------
    divisor(15): 4,327,390,717,482, ( 67
    digits)
    396,417,404,033,958,009,094,495,735,
    504,420,235,023,615,597,103,302,829
    ------------------------------------------------------------------------------
    a(15): 1 ( 1
    digit )
    ------------------------------------------------------------------------------
    p(15): 78,256,779 ( 8
    digits)
    ------------------------------------------------------------------------------
    q(15): 245,850,922 ( 9
    digits)
    ------------------------------------------------------------------------------
    dap_h(15): 318,309,886,183,790,679,458,993,446, ( 108
    digits)
    443,125,399,383,289,683,168,241,280,
    787,224,381,448,526,762,083,894,076,
    264,639,755,957,474,098,876,879,542
    ------------------------------------------------------------------------------
    dap_k(15): 1, ( 109
    digits)
    000,000,000,000,000,000,000,000,000,
    000,000,000,000,000,000,000,000,000,
    000,000,000,000,000,000,000,000,000,
    000,000,000,000,000,000,000,000,000
    ------------------------------------------------------------------------------
    dividend(16): 4,327,390,717,482, ( 67
    digits)
    396,417,404,033,958,009,094,495,735,
    504,420,235,023,615,597,103,302,829
    ------------------------------------------------------------------------------
    divisor(16): 3,059,032,691,500, ( 67
    digits)
    558,176,219,240,239,056,230,197,166,
    824,901,748,738,944,239,648,856,397
    ------------------------------------------------------------------------------
    a(16): 1 ( 1
    digit )
    ------------------------------------------------------------------------------
    p(16): 131,002,976 ( 9
    digits)
    ------------------------------------------------------------------------------
    q(16): 411,557,987 ( 9
    digits)
    ------------------------------------------------------------------------------
    dap_h(16): 318,309,886,183,790,669,575,803,907, ( 108
    digits)
    311,851,051,502,008,634,326,418,745,
    944,541,710,473,474,543,454,796,322,
    541,056,650,663,421,580,006,901,919
    ------------------------------------------------------------------------------
    dap_k(16): 1, ( 109
    digits)
    000,000,000,000,000,000,000,000,000,
    000,000,000,000,000,000,000,000,000,
    000,000,000,000,000,000,000,000,000,
    000,000,000,000,000,000,000,000,000
    ------------------------------------------------------------------------------
    dividend(17): 3,059,032,691,500, ( 67
    digits)
    558,176,219,240,239,056,230,197,166,
    824,901,748,738,944,239,648,856,397
    ------------------------------------------------------------------------------
    divisor(17): 1,268,358,025,981, ( 67
    digits)
    838,241,184,793,718,952,864,298,568,
    679,518,486,284,671,357,454,446,432
    ------------------------------------------------------------------------------
    a(17): 2 ( 1
    digit )
    ------------------------------------------------------------------------------
    p(17): 340,262,731 ( 9
    digits)
    ------------------------------------------------------------------------------
    q(17): 1,068,966,896 ( 10
    digits)
    ------------------------------------------------------------------------------
    dap_h(17): 318,309,886,183,790,671,848,831,509, ( 108
    digits)
    558,739,413,011,719,681,916,136,718,
    231,918,006,935,174,538,838,104,487,
    007,425,532,099,920,145,702,996,587
    ------------------------------------------------------------------------------
    dap_k(17): 1, ( 109
    digits)
    000,000,000,000,000,000,000,000,000,
    000,000,000,000,000,000,000,000,000,
    000,000,000,000,000,000,000,000,000,
    000,000,000,000,000,000,000,000,000
    ------------------------------------------------------------------------------
    dividend(18): 1,268,358,025,981, ( 67
    digits)
    838,241,184,793,718,952,864,298,568,
    679,518,486,284,671,357,454,446,432
    ------------------------------------------------------------------------------
    divisor(18): 522,316,639,536, ( 66
    digits)
    881,693,849,652,801,150,501,600,029,
    465,864,776,169,601,524,739,963,533
    ------------------------------------------------------------------------------
    a(18): 2 ( 1
    digit )
    ------------------------------------------------------------------------------
    p(18): 811,528,438 ( 9
    digits)
    ------------------------------------------------------------------------------
    q(18): 2,549,491,779 ( 10
    digits)
    ------------------------------------------------------------------------------
    dap_h(18): 318,309,886,183,790,671,481,902,432, ( 108
    digits)
    916,219,260,340,670,429,751,560,301,
    069,713,706,262,553,121,964,783,554,
    612,905,460,951,478,518,181,956,451
    ------------------------------------------------------------------------------
    dap_k(18): 1, ( 109
    digits)
    000,000,000,000,000,000,000,000,000,
    000,000,000,000,000,000,000,000,000,
    000,000,000,000,000,000,000,000,000,
    000,000,000,000,000,000,000,000,000
    ------------------------------------------------------------------------------
    dividend(19): 522,316,639,536, ( 66
    digits)
    881,693,849,652,801,150,501,600,029,
    465,864,776,169,601,524,739,963,533
    ------------------------------------------------------------------------------
    divisor(19): 223,724,746,908, ( 66
    digits)
    074,853,485,488,116,651,861,098,509,
    747,788,933,945,468,307,974,519,366
    ------------------------------------------------------------------------------
    a(19): 2 ( 1
    digit )
    ------------------------------------------------------------------------------
    p(19): 1,963,319,607 ( 10
    digits)
    ------------------------------------------------------------------------------
    q(19): 6,167,950,454 ( 10
    digits)
    ------------------------------------------------------------------------------
    dap_h(19): 318,309,886,183,790,671,545,494,875, ( 108
    digits)
    662,955,511,801,846,260,420,690,467,
    498,362,949,722,877,264,863,321,160,
    523,706,113,414,898,711,830,670,616
    ------------------------------------------------------------------------------
    dap_k(19): 1, ( 109
    digits)
    000,000,000,000,000,000,000,000,000,
    000,000,000,000,000,000,000,000,000,
    000,000,000,000,000,000,000,000,000,
    000,000,000,000,000,000,000,000,000
    ------------------------------------------------------------------------------
    dividend(20): 223,724,746,908, ( 66
    digits)
    074,853,485,488,116,651,861,098,509,
    747,788,933,945,468,307,974,519,366
    ------------------------------------------------------------------------------
    divisor(20): 74,867,145,720, ( 65
    digits)
    731,986,878,676,567,846,779,403,009,
    970,286,908,278,664,908,790,924,801
    ------------------------------------------------------------------------------
    a(20): 2 ( 1
    digit )
    ------------------------------------------------------------------------------
    p(20): 4,738,167,652 ( 10
    digits)
    ------------------------------------------------------------------------------
    q(20): 14,885,392,687 ( 11
    digits)
    ------------------------------------------------------------------------------
    dap_h(20): 318,309,886,183,790,671,534,603,096, ( 108
    digits)
    494,044,141,302,538,416,533,209,729,
    490,826,700,418,911,158,819,870,776,
    043,303,183,298,814,910,196,127,632
    ------------------------------------------------------------------------------
    dap_k(20): 1, ( 109
    digits)
    000,000,000,000,000,000,000,000,000,
    000,000,000,000,000,000,000,000,000,
    000,000,000,000,000,000,000,000,000,
    000,000,000,000,000,000,000,000,000
    ------------------------------------------------------------------------------
    dividend(21): 74,867,145,720, ( 65
    digits)
    731,986,878,676,567,846,779,403,009,
    970,286,908,278,664,908,790,924,801
    ------------------------------------------------------------------------------
    divisor(21): 73,990,455,466, ( 65
    digits)
    610,879,728,134,980,958,302,292,489,
    807,215,117,388,138,490,392,669,764
    ------------------------------------------------------------------------------
    a(21): 1 ( 1
    digit )
    ------------------------------------------------------------------------------
    p(21): 6,701,487,259 ( 10
    digits)
    ------------------------------------------------------------------------------
    q(21): 21,053,343,141 ( 11
    digits)
    ------------------------------------------------------------------------------
    dap_h(21): 318,309,886,183,790,671,537,794,036, ( 108
    digits)
    470,647,006,398,877,940,560,708,595,
    349,445,836,498,656,629,196,105,116,
    576,934,055,681,907,531,210,175,889
    ------------------------------------------------------------------------------
    dap_k(21): 1, ( 109
    digits)
    000,000,000,000,000,000,000,000,000,
    000,000,000,000,000,000,000,000,000,
    000,000,000,000,000,000,000,000,000,
    000,000,000,000,000,000,000,000,000
    ------------------------------------------------------------------------------
    dividend(22): 73,990,455,466, ( 65
    digits)
    610,879,728,134,980,958,302,292,489,
    807,215,117,388,138,490,392,669,764
    ------------------------------------------------------------------------------
    divisor(22): 876,690,254, ( 63
    digits)
    121,107,150,541,586,888,477,110,520,
    163,071,790,890,526,418,398,255,037
    ------------------------------------------------------------------------------
    a(22): 84 ( 2
    digits)
    ------------------------------------------------------------------------------
    p(22): 567,663,097,408 ( 12
    digits)
    ------------------------------------------------------------------------------
    q(22): 1,783,366,216,531 ( 13
    digits)
    ------------------------------------------------------------------------------
    dap_h(22): 318,309,886,183,790,671,537,767,402, ( 108
    digits)
    347,968,169,288,920,129,519,591,286,
    922,246,499,843,017,721,820,159,000,
    765,491,438,800,208,855,137,967,410
    ------------------------------------------------------------------------------
    dap_k(22): 1, ( 109
    digits)
    000,000,000,000,000,000,000,000,000,
    000,000,000,000,000,000,000,000,000,
    000,000,000,000,000,000,000,000,000,
    000,000,000,000,000,000,000,000,000
    ------------------------------------------------------------------------------
    dividend(23): 876,690,254, ( 63
    digits)
    121,107,150,541,586,888,477,110,520,
    163,071,790,890,526,418,398,255,037
    ------------------------------------------------------------------------------
    divisor(23): 348,474,120, ( 63
    digits)
    437,879,082,641,682,326,225,008,796,
    109,184,682,583,919,344,939,246,656
    ------------------------------------------------------------------------------
    a(23): 2 ( 1
    digit )
    ------------------------------------------------------------------------------
    p(23): 1,142,027,682,075 ( 13
    digits)
    ------------------------------------------------------------------------------
    q(23): 3,587,785,776,203 ( 13
    digits)
    ------------------------------------------------------------------------------
    dap_h(23): 318,309,886,183,790,671,537,767,558, ( 108
    digits)
    638,605,067,482,814,829,940,110,836,
    071,879,643,986,137,602,734,844,575,
    247,802,239,802,596,325,728,917,474
    ------------------------------------------------------------------------------
    dap_k(23): 1, ( 109
    digits)
    000,000,000,000,000,000,000,000,000,
    000,000,000,000,000,000,000,000,000,
    000,000,000,000,000,000,000,000,000,
    000,000,000,000,000,000,000,000,000
    ------------------------------------------------------------------------------
    dividend(24): 348,474,120, ( 63
    digits)
    437,879,082,641,682,326,225,008,796,
    109,184,682,583,919,344,939,246,656
    ------------------------------------------------------------------------------
    divisor(24): 179,742,013, ( 63
    digits)
    245,348,985,258,222,236,027,092,927,
    944,702,425,722,687,728,519,761,725
    ------------------------------------------------------------------------------
    a(24): 1 ( 1
    digit )
    ------------------------------------------------------------------------------
    p(24): 1,709,690,779,483 ( 13
    digits)
    ------------------------------------------------------------------------------
    q(24): 5,371,151,992,734 ( 13
    digits)
    ------------------------------------------------------------------------------
    dap_h(24): 318,309,886,183,790,671,537,767,506, ( 108
    digits)
    745,930,613,026,019,713,135,479,918,
    580,296,571,090,230,715,038,713,552,
    599,385,136,221,639,678,003,235,413
    ------------------------------------------------------------------------------
    dap_k(24): 1, ( 109
    digits)
    000,000,000,000,000,000,000,000,000,
    000,000,000,000,000,000,000,000,000,
    000,000,000,000,000,000,000,000,000,
    000,000,000,000,000,000,000,000,000
    ------------------------------------------------------------------------------
    dividend(25): 179,742,013, ( 63
    digits)
    245,348,985,258,222,236,027,092,927,
    944,702,425,722,687,728,519,761,725
    ------------------------------------------------------------------------------
    divisor(25): 168,732,107, ( 63
    digits)
    192,530,097,383,460,090,197,915,868,
    164,482,256,861,231,616,419,484,931
    ------------------------------------------------------------------------------
    a(25): 1 ( 1
    digit )
    ------------------------------------------------------------------------------
    p(25): 2,851,718,461,558 ( 13
    digits)
    ------------------------------------------------------------------------------
    q(25): 8,958,937,768,937 ( 13
    digits)
    ------------------------------------------------------------------------------
    dap_h(25): 318,309,886,183,790,671,537,767,527, ( 108
    digits)
    527,389,765,735,767,056,341,225,192,
    023,040,738,134,348,298,373,860,927,
    172,330,851,627,758,063,474,115,059
    ------------------------------------------------------------------------------
    dap_k(25): 1, ( 109
    digits)
    000,000,000,000,000,000,000,000,000,
    000,000,000,000,000,000,000,000,000,
    000,000,000,000,000,000,000,000,000,
    000,000,000,000,000,000,000,000,000
    ------------------------------------------------------------------------------
    dividend(26): 168,732,107, ( 63
    digits)
    192,530,097,383,460,090,197,915,868,
    164,482,256,861,231,616,419,484,931
    ------------------------------------------------------------------------------
    divisor(26): 11,009,906, ( 62
    digits)
    052,818,887,874,762,145,829,177,059,
    780,220,168,861,456,112,100,276,794
    ------------------------------------------------------------------------------
    a(26): 15 ( 2
    digits)
    ------------------------------------------------------------------------------
    p(26): 44,485,467,702,853 ( 14
    digits)
    ------------------------------------------------------------------------------
    q(26): 139,755,218,526,789 ( 15
    digits)
    ------------------------------------------------------------------------------
    dap_h(26): 318,309,886,183,790,671,537,767,526, ( 108
    digits)
    728,704,915,777,185,980,437,968,151,
    937,362,333,112,090,554,659,754,663,
    338,169,603,475,907,230,558,864,041
    ------------------------------------------------------------------------------
    dap_k(26): 1, ( 109
    digits)
    000,000,000,000,000,000,000,000,000,
    000,000,000,000,000,000,000,000,000,
    000,000,000,000,000,000,000,000,000,
    000,000,000,000,000,000,000,000,000
    ------------------------------------------------------------------------------
    dividend(27): 11,009,906, ( 62
    digits)
    052,818,887,874,762,145,829,177,059,
    780,220,168,861,456,112,100,276,794
    ------------------------------------------------------------------------------
    divisor(27): 3,583,516, ( 61
    digits)
    400,246,779,262,027,902,760,259,971,
    461,179,723,939,389,934,915,333,021
    ------------------------------------------------------------------------------
    a(27): 3 ( 1
    digit )
    ------------------------------------------------------------------------------
    p(27): 136,308,121,570,117 ( 15
    digits)
    ------------------------------------------------------------------------------
    q(27): 428,224,593,349,304 ( 15
    digits)
    ------------------------------------------------------------------------------
    dap_h(27): 318,309,886,183,790,671,537,767,526, ( 108
    digits)
    745,414,296,739,654,568,348,557,460,
    494,977,888,435,431,926,857,946,447,
    323,290,173,958,284,196,619,402,232
    ------------------------------------------------------------------------------
    dap_k(27): 1, ( 109
    digits)
    000,000,000,000,000,000,000,000,000,
    000,000,000,000,000,000,000,000,000,
    000,000,000,000,000,000,000,000,000,
    000,000,000,000,000,000,000,000,000
    ------------------------------------------------------------------------------
    dividend(28): 3,583,516, ( 61
    digits)
    400,246,779,262,027,902,760,259,971,
    461,179,723,939,389,934,915,333,021
    ------------------------------------------------------------------------------
    divisor(28): 259,356, ( 60
    digits)
    852,078,550,088,678,437,548,397,145,
    396,680,997,043,286,307,354,277,731
    ------------------------------------------------------------------------------
    a(28): 13 ( 2
    digits)
    ------------------------------------------------------------------------------
    p(28): 1,816,491,048,114,374 ( 16
    digits)
    ------------------------------------------------------------------------------
    q(28): 5,706,674,932,067,741 ( 16
    digits)
    ------------------------------------------------------------------------------
    dap_h(28): 318,309,886,183,790,671,537,767,526, ( 108
    digits)
    745,005,087,661,573,958,801,534,434,
    142,677,965,294,824,480,455,086,011,
    699,516,936,578,412,948,088,216,377
    ------------------------------------------------------------------------------
    dap_k(28): 1, ( 109
    digits)
    000,000,000,000,000,000,000,000,000,
    000,000,000,000,000,000,000,000,000,
    000,000,000,000,000,000,000,000,000,
    000,000,000,000,000,000,000,000,000
    ------------------------------------------------------------------------------
    dividend(29): 259,356, ( 60
    digits)
    852,078,550,088,678,437,548,397,145,
    396,680,997,043,286,307,354,277,731
    ------------------------------------------------------------------------------
    divisor(29): 211,877, ( 60
    digits)
    323,225,628,109,208,214,631,097,081,
    304,326,762,376,667,939,309,722,518
    ------------------------------------------------------------------------------
    a(29): 1 ( 1
    digit )
    ------------------------------------------------------------------------------
    p(29): 1,952,799,169,684,491 ( 16
    digits)
    ------------------------------------------------------------------------------
    q(29): 6,134,899,525,417,045 ( 16
    digits)
    ------------------------------------------------------------------------------
    dap_h(29): 318,309,886,183,790,671,537,767,526, ( 108
    digits)
    745,033,651,029,291,704,823,232,462,
    142,400,778,409,785,361,133,211,234,
    468,619,911,215,421,968,810,021,276
    ------------------------------------------------------------------------------
    dap_k(29): 1, ( 109
    digits)
    000,000,000,000,000,000,000,000,000,
    000,000,000,000,000,000,000,000,000,
    000,000,000,000,000,000,000,000,000,
    000,000,000,000,000,000,000,000,000
    ------------------------------------------------------------------------------
    dividend(30): 211,877, ( 60
    digits)
    323,225,628,109,208,214,631,097,081,
    304,326,762,376,667,939,309,722,518
    ------------------------------------------------------------------------------
    divisor(30): 47,479, ( 59
    digits)
    528,852,921,979,470,222,917,300,064,
    092,354,234,666,618,368,044,555,213
    ------------------------------------------------------------------------------
    a(30): 4 ( 1
    digit )
    ------------------------------------------------------------------------------
    p(30): 9,627,687,726,852,338 ( 16
    digits)
    ------------------------------------------------------------------------------
    q(30): 30,246,273,033,735,921 ( 17
    digits)
    ------------------------------------------------------------------------------
    dap_h(30): 318,309,886,183,790,671,537,767,526, ( 108
    digits)
    745,028,261,874,259,331,542,854,065,
    277,975,476,160,430,618,883,666,404,
    629,738,083,203,081,641,641,249,841
    ------------------------------------------------------------------------------
    dap_k(30): 1, ( 109
    digits)
    000,000,000,000,000,000,000,000,000,
    000,000,000,000,000,000,000,000,000,
    000,000,000,000,000,000,000,000,000,
    000,000,000,000,000,000,000,000,000
    ------------------------------------------------------------------------------
    dividend(31): 47,479, ( 59
    digits)
    528,852,921,979,470,222,917,300,064,
    092,354,234,666,618,368,044,555,213
    ------------------------------------------------------------------------------
    divisor(31): 21,959, ( 59
    digits)
    207,813,940,191,327,322,961,896,824,
    934,909,823,710,194,467,131,501,666
    ------------------------------------------------------------------------------
    a(31): 2 ( 1
    digit )
    ------------------------------------------------------------------------------
    p(31): 21,208,174,623,389,167 ( 17
    digits)
    ------------------------------------------------------------------------------
    q(31): 66,627,445,592,888,887 ( 17
    digits)
    ------------------------------------------------------------------------------
    dap_h(31): 318,309,886,183,790,671,537,767,526, ( 108
    digits)
    745,028,758,095,063,780,849,249,056,
    125,096,982,605,151,934,254,316,809,
    447,832,374,497,980,208,025,766,719
    ------------------------------------------------------------------------------
    dap_k(31): 1, ( 109
    digits)
    000,000,000,000,000,000,000,000,000,
    000,000,000,000,000,000,000,000,000,
    000,000,000,000,000,000,000,000,000,
    000,000,000,000,000,000,000,000,000
    ------------------------------------------------------------------------------
    dividend(32): 21,959, ( 59
    digits)
    207,813,940,191,327,322,961,896,824,
    934,909,823,710,194,467,131,501,666
    ------------------------------------------------------------------------------
    divisor(32): 3,561, ( 58
    digits)
    113,225,041,596,815,576,993,506,414,
    222,534,587,246,229,433,781,551,881
    ------------------------------------------------------------------------------
    a(32): 6 ( 1
    digit )
    ------------------------------------------------------------------------------
    p(32): 136,876,735,467,187,340 ( 18
    digits)
    ------------------------------------------------------------------------------
    q(32): 430,010,946,591,069,243 ( 18
    digits)
    ------------------------------------------------------------------------------
    dap_h(32): 318,309,886,183,790,671,537,767,526, ( 108
    digits)
    745,028,723,191,696,656,414,479,536,
    080,140,821,276,023,930,411,720,824,
    896,535,293,548,246,864,208,091,442
    ------------------------------------------------------------------------------
    dap_k(32): 1, ( 109
    digits)
    000,000,000,000,000,000,000,000,000,
    000,000,000,000,000,000,000,000,000,
    000,000,000,000,000,000,000,000,000,
    000,000,000,000,000,000,000,000,000
    ------------------------------------------------------------------------------
    dividend(33): 3,561, ( 58
    digits)
    113,225,041,596,815,576,993,506,414,
    222,534,587,246,229,433,781,551,881
    ------------------------------------------------------------------------------
    divisor(33): 592, ( 57
    digits)
    528,463,690,610,433,861,000,858,339,
    599,702,300,232,817,864,442,190,380
    ------------------------------------------------------------------------------
    a(33): 6 ( 1
    digit )
    ------------------------------------------------------------------------------
    p(33): 842,468,587,426,513,207 ( 18
    digits)
    ------------------------------------------------------------------------------
    q(33): 2,646,693,125,139,304,345 ( 19
    digits)
    ------------------------------------------------------------------------------
    dap_h(33): 318,309,886,183,790,671,537,767,526, ( 108
    digits)
    745,028,724,070,348,651,238,966,114,
    448,014,212,811,175,206,998,600,802,
    693,670,993,407,777,082,814,551,949
    ------------------------------------------------------------------------------
    dap_k(33): 1, ( 109
    digits)
    000,000,000,000,000,000,000,000,000,
    000,000,000,000,000,000,000,000,000,
    000,000,000,000,000,000,000,000,000,
    000,000,000,000,000,000,000,000,000
    ------------------------------------------------------------------------------
    dividend(34): 592, ( 57
    digits)
    528,463,690,610,433,861,000,858,339,
    599,702,300,232,817,864,442,190,380
    ------------------------------------------------------------------------------
    divisor(34): 5, ( 55
    digits)
    942,442,897,934,212,410,988,356,376,
    624,320,785,849,322,247,128,409,601
    ------------------------------------------------------------------------------
    a(34): 99 ( 2
    digits)
    ------------------------------------------------------------------------------
    p(34): 83,541,266,890,691,994,833 ( 20
    digits)
    ------------------------------------------------------------------------------
    q(34): 262,452,630,335,382,199,398 ( 21
    digits)
    ------------------------------------------------------------------------------
    dap_h(34): 318,309,886,183,790,671,537,767,526, ( 108
    digits)
    745,028,724,068,909,039,162,928,665,
    394,338,274,808,166,404,802,360,771,
    259,216,777,642,534,163,728,458,650
    ------------------------------------------------------------------------------
    dap_k(34): 1, ( 109
    digits)
    000,000,000,000,000,000,000,000,000,
    000,000,000,000,000,000,000,000,000,
    000,000,000,000,000,000,000,000,000,
    000,000,000,000,000,000,000,000,000
    ------------------------------------------------------------------------------
    dividend(35): 5, ( 55
    digits)
    942,442,897,934,212,410,988,356,376,
    624,320,785,849,322,247,128,409,601
    ------------------------------------------------------------------------------
    divisor(35): 4, ( 55
    digits)
    226,616,795,123,405,173,153,577,053,
    791,944,501,149,915,398,729,639,881
    ------------------------------------------------------------------------------
    a(35): 1 ( 1
    digit )
    ------------------------------------------------------------------------------
    p(35): 84,383,735,478,118,508,040 ( 20
    digits)
    ------------------------------------------------------------------------------
    q(35): 265,099,323,460,521,503,743 ( 21
    digits)
    ------------------------------------------------------------------------------
    dap_h(35): 318,309,886,183,790,671,537,767,526, ( 108
    digits)
    745,028,724,068,923,411,932,103,548,
    834,522,842,872,513,504,413,841,039,
    138,614,687,723,361,129,035,291,318
    ------------------------------------------------------------------------------
    dap_k(35): 1, ( 109
    digits)
    000,000,000,000,000,000,000,000,000,
    000,000,000,000,000,000,000,000,000,
    000,000,000,000,000,000,000,000,000,
    000,000,000,000,000,000,000,000,000
    ------------------------------------------------------------------------------
    dividend(36): 4, ( 55
    digits)
    226,616,795,123,405,173,153,577,053,
    791,944,501,149,915,398,729,639,881
    ------------------------------------------------------------------------------
    divisor(36): 1, ( 55
    digits)
    715,826,102,810,807,237,834,779,322,
    832,376,284,699,406,848,398,769,720
    ------------------------------------------------------------------------------
    a(36): 2 ( 1
    digit )
    ------------------------------------------------------------------------------
    p(36): 252,308,737,846,929,010,913 ( 21
    digits)
    ------------------------------------------------------------------------------
    q(36): 792,651,277,256,425,206,884 ( 21
    digits)
    ------------------------------------------------------------------------------
    dap_h(36): 318,309,886,183,790,671,537,767,526, ( 108
    digits)
    745,028,724,068,918,653,003,198,257,
    800,180,493,281,310,073,859,628,554,
    285,406,680,284,665,418,164,625,029
    ------------------------------------------------------------------------------
    dap_k(36): 1, ( 109
    digits)
    000,000,000,000,000,000,000,000,000,
    000,000,000,000,000,000,000,000,000,
    000,000,000,000,000,000,000,000,000,
    000,000,000,000,000,000,000,000,000
    ------------------------------------------------------------------------------
    dividend(37): 1, ( 55
    digits)
    715,826,102,810,807,237,834,779,322,
    832,376,284,699,406,848,398,769,720
    ------------------------------------------------------------------------------
    divisor(37): 794,964,589,501,790,697,484,018,408, ( 54
    digits)
    127,191,931,751,101,701,932,100,441
    ------------------------------------------------------------------------------
    a(37): 2 ( 1
    digit )
    ------------------------------------------------------------------------------
    p(37): 589,001,211,171,976,529,866 ( 21
    digits)
    ------------------------------------------------------------------------------
    q(37): 1,850,401,877,973,371,917,511 ( 22
    digits)
    ------------------------------------------------------------------------------
    dap_h(37): 318,309,886,183,790,671,537,767,526, ( 108
    digits)
    745,028,724,068,919,334,795,001,544,
    938,989,901,437,788,026,049,454,430,
    600,908,281,920,522,084,321,675,623
    ------------------------------------------------------------------------------
    dap_k(37): 1, ( 109
    digits)
    000,000,000,000,000,000,000,000,000,
    000,000,000,000,000,000,000,000,000,
    000,000,000,000,000,000,000,000,000,
    000,000,000,000,000,000,000,000,000
    ------------------------------------------------------------------------------
    dividend(38): 794,964,589,501,790,697,484,018,408, ( 54
    digits)
    127,191,931,751,101,701,932,100,441
    ------------------------------------------------------------------------------
    divisor(38): 125,896,923,807,225,842,866,742,506, ( 54
    digits)
    577,992,421,197,203,444,534,568,838
    ------------------------------------------------------------------------------
    a(38): 6 ( 1
    digit )
    ------------------------------------------------------------------------------
    p(38): 3,786,316,004,878,788,190,109 ( 22
    digits)
    ------------------------------------------------------------------------------
    q(38): 11,895,062,545,096,656,711,950 ( 23
    digits)
    ------------------------------------------------------------------------------
    dap_h(38): 318,309,886,183,790,671,537,767,526, ( 108
    digits)
    745,028,724,068,919,289,362,441,468,
    045,285,994,406,211,981,971,828,249,
    614,967,485,138,355,225,805,192,779
    ------------------------------------------------------------------------------
    dap_k(38): 1, ( 109
    digits)
    000,000,000,000,000,000,000,000,000,
    000,000,000,000,000,000,000,000,000,
    000,000,000,000,000,000,000,000,000,
    000,000,000,000,000,000,000,000,000
    ------------------------------------------------------------------------------
    dividend(39): 125,896,923,807,225,842,866,742,506, ( 54
    digits)
    577,992,421,197,203,444,534,568,838
    ------------------------------------------------------------------------------
    divisor(39): 39,583,046,658,435,640,283,563,368, ( 53
    digits)
    659,237,404,567,881,034,724,687,413
    ------------------------------------------------------------------------------
    a(39): 3 ( 1
    digit )
    ------------------------------------------------------------------------------
    p(39): 11,947,949,225,808,341,100,193 ( 23
    digits)
    ------------------------------------------------------------------------------
    q(39): 37,535,589,513,263,342,053,361 ( 23
    digits)
    ------------------------------------------------------------------------------
    dap_h(39): 318,309,886,183,790,671,537,767,526, ( 108
    digits)
    745,028,724,068,919,291,602,142,391,
    290,842,926,882,227,563,012,147,516,
    387,901,470,038,334,006,146,460,351
    ------------------------------------------------------------------------------
    dap_k(39): 1, ( 109
    digits)
    000,000,000,000,000,000,000,000,000,
    000,000,000,000,000,000,000,000,000,
    000,000,000,000,000,000,000,000,000,
    000,000,000,000,000,000,000,000,000
    ------------------------------------------------------------------------------
    dividend(40): 39,583,046,658,435,640,283,563,368, ( 53
    digits)
    659,237,404,567,881,034,724,687,413
    ------------------------------------------------------------------------------
    divisor(40): 7,147,783,831,918,922,016,052,400, ( 52
    digits)
    600,280,207,493,560,340,360,506,599
    ------------------------------------------------------------------------------
    a(40): 5 ( 1
    digit )
    ------------------------------------------------------------------------------
    p(40): 63,526,062,133,920,493,691,074 ( 23
    digits)
    ------------------------------------------------------------------------------
    q(40): 199,573,010,111,413,366,978,755 ( 24
    digits)
    ------------------------------------------------------------------------------
    dap_h(40): 318,309,886,183,790,671,537,767,526, ( 108
    digits)
    745,028,724,068,919,291,468,650,479,
    987,586,825,621,702,619,006,350,232,
    699,889,621,018,702,416,161,692,148
    ------------------------------------------------------------------------------
    dap_k(40): 1, ( 109
    digits)
    000,000,000,000,000,000,000,000,000,
    000,000,000,000,000,000,000,000,000,
    000,000,000,000,000,000,000,000,000,
    000,000,000,000,000,000,000,000,000
    ------------------------------------------------------------------------------
    dividend(41): 7,147,783,831,918,922,016,052,400, ( 52
    digits)
    600,280,207,493,560,340,360,506,599
    ------------------------------------------------------------------------------
    divisor(41): 3,844,127,498,841,030,203,301,365, ( 52
    digits)
    657,836,367,100,079,332,922,154,418
    ------------------------------------------------------------------------------
    a(41): 1 ( 1
    digit )
    ------------------------------------------------------------------------------
    p(41): 75,474,011,359,728,834,791,267 ( 23
    digits)
    ------------------------------------------------------------------------------
    q(41): 237,108,599,624,676,709,032,116 ( 24
    digits)
    ------------------------------------------------------------------------------
    dap_h(41): 318,309,886,183,790,671,537,767,526, ( 108
    digits)
    745,028,724,068,919,291,489,782,980,
    428,108,592,959,430,555,515,421,077,
    137,315,846,835,622,913,835,499,396
    ------------------------------------------------------------------------------
    dap_k(41): 1, ( 109
    digits)
    000,000,000,000,000,000,000,000,000,
    000,000,000,000,000,000,000,000,000,
    000,000,000,000,000,000,000,000,000,
    000,000,000,000,000,000,000,000,000
    ------------------------------------------------------------------------------
    dividend(42): 3,844,127,498,841,030,203,301,365, ( 52
    digits)
    657,836,367,100,079,332,922,154,418
    ------------------------------------------------------------------------------
    divisor(42): 3,303,656,333,077,891,812,751,034, ( 52
    digits)
    942,443,840,393,481,007,438,352,181
    ------------------------------------------------------------------------------
    a(42): 1 ( 1
    digit )
    ------------------------------------------------------------------------------
    p(42): 139,000,073,493,649,328,482,341 ( 24
    digits)
    ------------------------------------------------------------------------------
    q(42): 436,681,609,736,090,076,010,871 ( 24
    digits)
    ------------------------------------------------------------------------------
    dap_h(42): 318,309,886,183,790,671,537,767,526, ( 108
    digits)
    745,028,724,068,919,291,480,124,967,
    289,784,440,453,695,907,041,082,324,
    703,691,948,430,919,860,989,708,599
    ------------------------------------------------------------------------------
    dap_k(42): 1, ( 109
    digits)
    000,000,000,000,000,000,000,000,000,
    000,000,000,000,000,000,000,000,000,
    000,000,000,000,000,000,000,000,000,
    000,000,000,000,000,000,000,000,000
    ------------------------------------------------------------------------------
    dividend(43): 3,303,656,333,077,891,812,751,034, ( 52
    digits)
    942,443,840,393,481,007,438,352,181
    ------------------------------------------------------------------------------
    divisor(43): 540,471,165,763,138,390,550,330, ( 51
    digits)
    715,392,526,706,598,325,483,802,237
    ------------------------------------------------------------------------------
    a(43): 6 ( 1
    digit )
    ------------------------------------------------------------------------------
    p(43): 909,474,452,321,624,805,685,313 ( 24
    digits)
    ------------------------------------------------------------------------------
    q(43): 2,857,198,258,041,217,165,097,342 ( 25
    digits)
    ------------------------------------------------------------------------------
    dap_h(43): 318,309,886,183,790,671,537,767,526, ( 108
    digits)
    745,028,724,068,919,291,480,926,451,
    038,397,216,621,622,908,041,821,354,
    131,274,430,180,490,770,793,800,202
    ------------------------------------------------------------------------------
    dap_k(43): 1, ( 109
    digits)
    000,000,000,000,000,000,000,000,000,
    000,000,000,000,000,000,000,000,000,
    000,000,000,000,000,000,000,000,000,
    000,000,000,000,000,000,000,000,000
    ------------------------------------------------------------------------------
    dividend(44): 540,471,165,763,138,390,550,330, ( 51
    digits)
    715,392,526,706,598,325,483,802,237
    ------------------------------------------------------------------------------
    divisor(44): 60,829,338,499,061,469,449,050, ( 50
    digits)
    650,088,680,153,891,054,535,538,759
    ------------------------------------------------------------------------------
    a(44): 8 ( 1
    digit )
    ------------------------------------------------------------------------------
    p(44): 7,414,795,692,066,647,773,964,845 ( 25
    digits)
    ------------------------------------------------------------------------------
    q(44): 23,294,267,674,065,827,396,789,607 ( 26
    digits)
    ------------------------------------------------------------------------------
    dap_h(44): 318,309,886,183,790,671,537,767,526, ( 108
    digits)
    745,028,724,068,919,291,480,911,426,
    173,979,126,317,132,947,129,077,921,
    486,530,383,104,229,946,930,855,009
    ------------------------------------------------------------------------------
    dap_k(44): 1, ( 109
    digits)
    000,000,000,000,000,000,000,000,000,
    000,000,000,000,000,000,000,000,000,
    000,000,000,000,000,000,000,000,000,
    000,000,000,000,000,000,000,000,000
    ------------------------------------------------------------------------------
    dividend(45): 60,829,338,499,061,469,449,050, ( 50
    digits)
    650,088,680,153,891,054,535,538,759
    ------------------------------------------------------------------------------
    divisor(45): 53,836,457,770,646,634,957,925, ( 50
    digits)
    514,683,085,475,469,889,199,492,165
    ------------------------------------------------------------------------------
    a(45): 1 ( 1
    digit )
    ------------------------------------------------------------------------------
    p(45): 8,324,270,144,388,272,579,650,158 ( 25
    digits)
    ------------------------------------------------------------------------------
    q(45): 26,151,465,932,107,044,561,886,949 ( 26
    digits)
    ------------------------------------------------------------------------------
    dap_h(45): 318,309,886,183,790,671,537,767,526, ( 108
    digits)
    745,028,724,068,919,291,480,913,067,
    726,944,230,566,596,507,476,434,894,
    534,547,792,904,191,355,711,666,087
    ------------------------------------------------------------------------------
    dap_k(45): 1, ( 109
    digits)
    000,000,000,000,000,000,000,000,000,
    000,000,000,000,000,000,000,000,000,
    000,000,000,000,000,000,000,000,000,
    000,000,000,000,000,000,000,000,000
    ------------------------------------------------------------------------------
    dividend(46): 53,836,457,770,646,634,957,925, ( 50
    digits)
    514,683,085,475,469,889,199,492,165
    ------------------------------------------------------------------------------
    divisor(46): 6,992,880,728,414,834,491,125, ( 49
    digits)
    135,405,594,678,421,165,336,046,594
    ------------------------------------------------------------------------------
    a(46): 7 ( 1
    digit )
    ------------------------------------------------------------------------------
    p(46): 65,684,686,702,784,555,831,515,951 ( 26
    digits)
    ------------------------------------------------------------------------------
    q(46): 206,354,529,198,815,139,329,998,250 ( 27
    digits)
    ------------------------------------------------------------------------------
    dap_h(46): 318,309,886,183,790,671,537,767,526, ( 108
    digits)
    745,028,724,068,919,291,480,912,882,
    420,741,750,982,787,553,424,394,016,
    727,413,357,725,429,207,046,116,895
    ------------------------------------------------------------------------------
    dap_k(46): 1, ( 109
    digits)
    000,000,000,000,000,000,000,000,000,
    000,000,000,000,000,000,000,000,000,
    000,000,000,000,000,000,000,000,000,
    000,000,000,000,000,000,000,000,000
    ------------------------------------------------------------------------------
    dividend(47): 6,992,880,728,414,834,491,125, ( 49
    digits)
    135,405,594,678,421,165,336,046,594
    ------------------------------------------------------------------------------
    divisor(47): 4,886,292,671,742,793,520,049, ( 49
    digits)
    566,843,922,726,521,731,847,166,007
    ------------------------------------------------------------------------------
    a(47): 1 ( 1
    digit )
    ------------------------------------------------------------------------------
    p(47): 74,008,956,847,172,828,411,166,109 ( 26
    digits)
    ------------------------------------------------------------------------------
    q(47): 232,505,995,130,922,183,891,885,199 ( 27
    digits)
    ------------------------------------------------------------------------------
    dap_h(47): 318,309,886,183,790,671,537,767,526, ( 108
    digits)
    745,028,724,068,919,291,480,912,903,
    263,339,114,651,657,948,913,647,155,
    994,171,549,850,667,282,266,900,601
    ------------------------------------------------------------------------------
    dap_k(47): 1, ( 109
    digits)
    000,000,000,000,000,000,000,000,000,
    000,000,000,000,000,000,000,000,000,
    000,000,000,000,000,000,000,000,000,
    000,000,000,000,000,000,000,000,000
    ------------------------------------------------------------------------------
    dividend(48): 4,886,292,671,742,793,520,049, ( 49
    digits)
    566,843,922,726,521,731,847,166,007
    ------------------------------------------------------------------------------
    divisor(48): 2,106,588,056,672,040,971,075, ( 49
    digits)
    568,561,671,951,899,433,488,880,587
    ------------------------------------------------------------------------------
    a(48): 2 ( 1
    digit )
    ------------------------------------------------------------------------------
    p(48): 213,702,600,397,130,212,653,848,169 ( 27
    digits)
    ------------------------------------------------------------------------------
    q(48): 671,366,519,460,659,507,113,768,648 ( 27
    digits)
    ------------------------------------------------------------------------------
    dap_h(48): 318,309,886,183,790,671,537,767,526, ( 108
    digits)
    745,028,724,068,919,291,480,912,896,
    857,055,450,528,038,036,711,956,528,
    864,997,557,136,633,583,077,875,075
    ------------------------------------------------------------------------------
    dap_k(48): 1, ( 109
    digits)
    000,000,000,000,000,000,000,000,000,
    000,000,000,000,000,000,000,000,000,
    000,000,000,000,000,000,000,000,000,
    000,000,000,000,000,000,000,000,000
    ------------------------------------------------------------------------------
    dividend(49): 2,106,588,056,672,040,971,075, ( 49
    digits)
    568,561,671,951,899,433,488,880,587
    ------------------------------------------------------------------------------
    divisor(49): 673,116,558,398,711,577,898, ( 48
    digits)
    429,720,578,822,722,864,869,404,833
    ------------------------------------------------------------------------------
    a(49): 3 ( 1
    digit )
    ------------------------------------------------------------------------------
    p(49): 715,116,758,038,563,466,372,710,616 ( 27
    digits)
    ------------------------------------------------------------------------------
    q(49): 2, ( 28
    digits)
    246,605,553,512,900,705,233,191,143
    ------------------------------------------------------------------------------
    dap_h(49): 318,309,886,183,790,671,537,767,526, ( 108
    digits)
    745,028,724,068,919,291,480,912,897,
    520,055,395,529,055,115,238,536,134,
    920,503,831,407,928,291,080,280,230
    ------------------------------------------------------------------------------
    dap_k(49): 1, ( 109
    digits)
    000,000,000,000,000,000,000,000,000,
    000,000,000,000,000,000,000,000,000,
    000,000,000,000,000,000,000,000,000,
    000,000,000,000,000,000,000,000,000
    ------------------------------------------------------------------------------
    dividend(50): 673,116,558,398,711,577,898, ( 48
    digits)
    429,720,578,822,722,864,869,404,833
    ------------------------------------------------------------------------------
    divisor(50): 87,238,381,475,906,237,380, ( 47
    digits)
    279,399,935,483,730,838,880,666,088
    ------------------------------------------------------------------------------
    a(50): 7 ( 1
    digit )
    ------------------------------------------------------------------------------
    p(50): 5, ( 28
    digits)
    219,519,906,667,074,477,262,822,481
    ------------------------------------------------------------------------------
    q(50): 16, ( 29
    digits)
    397,605,394,050,964,443,746,106,649
    ------------------------------------------------------------------------------
    dap_h(50): 318,309,886,183,790,671,537,767,526, ( 108
    digits)
    745,028,724,068,919,291,480,912,897,
    492,910,214,584,560,063,420,158,412,
    826,981,456,232,795,777,587,442,346
    ------------------------------------------------------------------------------
    dap_k(50): 1, ( 109
    digits)
    000,000,000,000,000,000,000,000,000,
    000,000,000,000,000,000,000,000,000,
    000,000,000,000,000,000,000,000,000,
    000,000,000,000,000,000,000,000,000
    ------------------------------------------------------------------------------
    dividend(51): 87,238,381,475,906,237,380, ( 47
    digits)
    279,399,935,483,730,838,880,666,088
    ------------------------------------------------------------------------------
    divisor(51): 62,447,888,067,367,916,236, ( 47
    digits)
    473,921,030,436,606,992,704,742,217
    ------------------------------------------------------------------------------
    a(51): 1 ( 1
    digit )
    ------------------------------------------------------------------------------
    p(51): 5, ( 28
    digits)
    934,636,664,705,637,943,635,533,097
    ------------------------------------------------------------------------------
    q(51): 18, ( 29
    digits)
    644,210,947,563,865,148,979,297,792
    ------------------------------------------------------------------------------
    dap_h(51): 318,309,886,183,790,671,537,767,526, ( 108
    digits)
    745,028,724,068,919,291,480,912,897,
    496,181,177,052,885,045,353,050,103,
    255,472,691,854,454,363,904,171,357
    ------------------------------------------------------------------------------
    dap_k(51): 1, ( 109
    digits)
    000,000,000,000,000,000,000,000,000,
    000,000,000,000,000,000,000,000,000,
    000,000,000,000,000,000,000,000,000,
    000,000,000,000,000,000,000,000,000
    ------------------------------------------------------------------------------
    dividend(52): 62,447,888,067,367,916,236, ( 47
    digits)
    473,921,030,436,606,992,704,742,217
    ------------------------------------------------------------------------------
    divisor(52): 24,790,493,408,538,321,143, ( 47
    digits)
    805,478,905,047,123,846,175,923,871
    ------------------------------------------------------------------------------
    a(52): 2 ( 1
    digit )
    ------------------------------------------------------------------------------
    p(52): 17, ( 29
    digits)
    088,793,236,078,350,364,533,888,675
    ------------------------------------------------------------------------------
    q(52): 53, ( 29
    digits)
    686,027,289,178,694,741,704,702,233
    ------------------------------------------------------------------------------
    dap_h(52): 318,309,886,183,790,671,537,767,526, ( 108
    digits)
    745,028,724,068,919,291,480,912,897,
    495,182,109,799,772,144,738,220,084,
    452,772,678,243,971,876,575,298,211
    ------------------------------------------------------------------------------
    dap_k(52): 1, ( 109
    digits)
    000,000,000,000,000,000,000,000,000,
    000,000,000,000,000,000,000,000,000,
    000,000,000,000,000,000,000,000,000,
    000,000,000,000,000,000,000,000,000
    ------------------------------------------------------------------------------
    dividend(53): 24,790,493,408,538,321,143, ( 47
    digits)
    805,478,905,047,123,846,175,923,871
    ------------------------------------------------------------------------------
    divisor(53): 12,866,901,250,291,273,948, ( 47
    digits)
    862,963,220,342,359,300,352,894,475
    ------------------------------------------------------------------------------
    a(53): 1 ( 1
    digit )
    ------------------------------------------------------------------------------
    p(53): 23, ( 29
    digits)
    023,429,900,783,988,308,169,421,772
    ------------------------------------------------------------------------------
    q(53): 72, ( 29
    digits)
    330,238,236,742,559,890,684,000,025
    ------------------------------------------------------------------------------
    dap_h(53): 318,309,886,183,790,671,537,767,526, ( 108
    digits)
    745,028,724,068,919,291,480,912,897,
    495,439,634,468,180,854,404,589,165,
    452,600,233,955,754,574,002,887,332
    ------------------------------------------------------------------------------
    dap_k(53): 1, ( 109
    digits)
    000,000,000,000,000,000,000,000,000,
    000,000,000,000,000,000,000,000,000,
    000,000,000,000,000,000,000,000,000,
    000,000,000,000,000,000,000,000,000
    ------------------------------------------------------------------------------
    dividend(54): 12,866,901,250,291,273,948, ( 47
    digits)
    862,963,220,342,359,300,352,894,475
    ------------------------------------------------------------------------------
    divisor(54): 11,923,592,158,247,047,194, ( 47
    digits)
    942,515,684,704,764,545,823,029,396
    ------------------------------------------------------------------------------
    a(54): 1 ( 1
    digit )
    ------------------------------------------------------------------------------
    p(54): 40, ( 29
    digits)
    112,223,136,862,338,672,703,310,447
    ------------------------------------------------------------------------------
    q(54): 126, ( 30
    digits)
    016,265,525,921,254,632,388,702,258
    ------------------------------------------------------------------------------
    dap_h(54): 318,309,886,183,790,671,537,767,526, ( 108
    digits)
    745,028,724,068,919,291,480,912,897,
    495,329,922,628,061,430,986,870,530,
    086,217,649,168,718,046,629,679,298
    ------------------------------------------------------------------------------
    dap_k(54): 1, ( 109
    digits)
    000,000,000,000,000,000,000,000,000,
    000,000,000,000,000,000,000,000,000,
    000,000,000,000,000,000,000,000,000,
    000,000,000,000,000,000,000,000,000
    ------------------------------------------------------------------------------
    dividend(55): 11,923,592,158,247,047,194, ( 47
    digits)
    942,515,684,704,764,545,823,029,396
    ------------------------------------------------------------------------------
    divisor(55): 943,309,092,044,226,753, ( 45
    digits)
    920,447,535,637,594,754,529,865,079
    ------------------------------------------------------------------------------
    a(55): 12 ( 2
    digits)
    ------------------------------------------------------------------------------
    p(55): 504, ( 30
    digits)
    370,107,543,132,052,380,609,147,136
    ------------------------------------------------------------------------------
    q(55): 1,584, ( 31
    digits)
    525,424,547,797,615,479,348,427,121
    ------------------------------------------------------------------------------
    dap_h(55): 318,309,886,183,790,671,537,767,526, ( 108
    digits)
    745,028,724,068,919,291,480,912,897,
    495,334,930,741,790,810,865,440,120,
    254,472,433,904,050,055,307,112,437
    ------------------------------------------------------------------------------
    dap_k(55): 1, ( 109
    digits)
    000,000,000,000,000,000,000,000,000,
    000,000,000,000,000,000,000,000,000,
    000,000,000,000,000,000,000,000,000,
    000,000,000,000,000,000,000,000,000
    ------------------------------------------------------------------------------
    dividend(56): 943,309,092,044,226,753, ( 45
    digits)
    920,447,535,637,594,754,529,865,079
    ------------------------------------------------------------------------------
    divisor(56): 603,883,053,716,326,147, ( 45
    digits)
    897,145,257,053,627,491,464,648,448
    ------------------------------------------------------------------------------
    a(56): 1 ( 1
    digit )
    ------------------------------------------------------------------------------
    p(56): 544, ( 30
    digits)
    482,330,679,994,391,053,312,457,583
    ------------------------------------------------------------------------------
    q(56): 1,710, ( 31
    digits)
    541,690,073,718,870,111,737,129,379
    ------------------------------------------------------------------------------
    dap_h(56): 318,309,886,183,790,671,537,767,526, ( 108
    digits)
    745,028,724,068,919,291,480,912,897,
    495,334,561,792,122,312,374,785,560,
    035,129,661,821,384,203,835,464,051
    ------------------------------------------------------------------------------
    dap_k(56): 1, ( 109
    digits)
    000,000,000,000,000,000,000,000,000,
    000,000,000,000,000,000,000,000,000,
    000,000,000,000,000,000,000,000,000,
    000,000,000,000,000,000,000,000,000
    ------------------------------------------------------------------------------
    dividend(57): 603,883,053,716,326,147, ( 45
    digits)
    897,145,257,053,627,491,464,648,448
    ------------------------------------------------------------------------------
    divisor(57): 339,426,038,327,900,606, ( 45
    digits)
    023,302,278,583,967,263,065,216,631
    ------------------------------------------------------------------------------
    a(57): 1 ( 1
    digit )
    ------------------------------------------------------------------------------
    p(57): 1,048, ( 31
    digits)
    852,438,223,126,443,433,921,604,719
    ------------------------------------------------------------------------------
    q(57): 3,295, ( 31
    digits)
    067,114,621,516,485,591,085,556,500
    ------------------------------------------------------------------------------
    dap_h(57): 318,309,886,183,790,671,537,767,526, ( 108
    digits)
    745,028,724,068,919,291,480,912,897,
    495,334,739,211,916,743,933,902,295,
    841,582,604,176,137,829,422,417,951
    ------------------------------------------------------------------------------
    dap_k(57): 1, ( 109
    digits)
    000,000,000,000,000,000,000,000,000,
    000,000,000,000,000,000,000,000,000,
    000,000,000,000,000,000,000,000,000,
    000,000,000,000,000,000,000,000,000
    ------------------------------------------------------------------------------
    dividend(58): 339,426,038,327,900,606, ( 45
    digits)
    023,302,278,583,967,263,065,216,631
    ------------------------------------------------------------------------------
    divisor(58): 264,457,015,388,425,541, ( 45
    digits)
    873,842,978,469,660,228,399,431,817
    ------------------------------------------------------------------------------
    a(58): 1 ( 1
    digit )
    ------------------------------------------------------------------------------
    p(58): 1,593, ( 31
    digits)
    334,768,903,120,834,487,234,062,302
    ------------------------------------------------------------------------------
    q(58): 5,005, ( 31
    digits)
    608,804,695,235,355,702,822,685,879
    ------------------------------------------------------------------------------
    dap_h(58): 318,309,886,183,790,671,537,767,526, ( 108
    digits)
    745,028,724,068,919,291,480,912,897,
    495,334,678,583,136,737,229,319,988,
    786,926,518,545,195,086,174,446,354
    ------------------------------------------------------------------------------
    dap_k(58): 1, ( 109
    digits)
    000,000,000,000,000,000,000,000,000,
    000,000,000,000,000,000,000,000,000,
    000,000,000,000,000,000,000,000,000,
    000,000,000,000,000,000,000,000,000
    ------------------------------------------------------------------------------
    dividend(59): 264,457,015,388,425,541, ( 45
    digits)
    873,842,978,469,660,228,399,431,817
    ------------------------------------------------------------------------------
    divisor(59): 74,969,022,939,475,064, ( 44
    digits)
    149,459,300,114,307,034,665,784,814
    ------------------------------------------------------------------------------
    a(59): 3 ( 1
    digit )
    ------------------------------------------------------------------------------
    p(59): 5,828, ( 31
    digits)
    856,744,932,488,946,895,623,791,625
    ------------------------------------------------------------------------------
    q(59): 18,311, ( 32
    digits)
    893,528,707,222,552,699,553,614,137
    ------------------------------------------------------------------------------
    dap_h(59): 318,309,886,183,790,671,537,767,526, ( 108
    digits)
    745,028,724,068,919,291,480,912,897,
    495,334,689,492,762,161,084,570,672,
    683,830,623,894,970,618,859,873,687
    ------------------------------------------------------------------------------
    dap_k(59): 1, ( 109
    digits)
    000,000,000,000,000,000,000,000,000,
    000,000,000,000,000,000,000,000,000,
    000,000,000,000,000,000,000,000,000,
    000,000,000,000,000,000,000,000,000
    ------------------------------------------------------------------------------
    dividend(60): 74,969,022,939,475,064, ( 44
    digits)
    149,459,300,114,307,034,665,784,814
    ------------------------------------------------------------------------------
    divisor(60): 39,549,946,570,000,349, ( 44
    digits)
    425,465,078,126,739,124,402,077,375
    ------------------------------------------------------------------------------
    a(60): 1 ( 1
    digit )
    ------------------------------------------------------------------------------
    p(60): 7,422, ( 31
    digits)
    191,513,835,609,781,382,857,853,927
    ------------------------------------------------------------------------------
    q(60): 23,317, ( 32
    digits)
    502,333,402,457,908,402,376,300,016
    ------------------------------------------------------------------------------
    dap_h(60): 318,309,886,183,790,671,537,767,526, ( 108
    digits)
    745,028,724,068,919,291,480,912,897,
    495,334,687,150,773,885,635,846,176,
    817,391,413,730,499,367,693,394,860
    ------------------------------------------------------------------------------
    dap_k(60): 1, ( 109
    digits)
    000,000,000,000,000,000,000,000,000,
    000,000,000,000,000,000,000,000,000,
    000,000,000,000,000,000,000,000,000,
    000,000,000,000,000,000,000,000,000
    ------------------------------------------------------------------------------
    dividend(61): 39,549,946,570,000,349, ( 44
    digits)
    425,465,078,126,739,124,402,077,375
    ------------------------------------------------------------------------------
    divisor(61): 35,419,076,369,474,714, ( 44
    digits)
    723,994,221,987,567,910,263,707,439
    ------------------------------------------------------------------------------
    a(61): 1 ( 1
    digit )
    ------------------------------------------------------------------------------
    p(61): 13,251, ( 32
    digits)
    048,258,768,098,728,278,481,645,552
    ------------------------------------------------------------------------------
    q(61): 41,629, ( 32
    digits)
    395,862,109,680,461,101,929,914,153
    ------------------------------------------------------------------------------
    dap_h(61): 318,309,886,183,790,671,537,767,526, ( 108
    digits)
    745,028,724,068,919,291,480,912,897,
    495,334,688,180,965,149,702,014,441,
    822,539,371,300,627,765,905,265,380
    ------------------------------------------------------------------------------
    dap_k(61): 1, ( 109
    digits)
    000,000,000,000,000,000,000,000,000,
    000,000,000,000,000,000,000,000,000,
    000,000,000,000,000,000,000,000,000,
    000,000,000,000,000,000,000,000,000
    ------------------------------------------------------------------------------
    dividend(62): 35,419,076,369,474,714, ( 44
    digits)
    723,994,221,987,567,910,263,707,439
    ------------------------------------------------------------------------------
    divisor(62): 4,130,870,200,525,634, ( 43
    digits)
    701,470,856,139,171,214,138,369,936
    ------------------------------------------------------------------------------
    a(62): 8 ( 1
    digit )
    ------------------------------------------------------------------------------
    p(62): 113,430, ( 33
    digits)
    577,583,980,399,607,610,711,018,343
    ------------------------------------------------------------------------------
    q(62): 356,352, ( 33
    digits)
    669,230,279,901,597,217,815,613,240
    ------------------------------------------------------------------------------
    dap_h(62): 318,309,886,183,790,671,537,767,526, ( 108
    digits)
    745,028,724,068,919,291,480,912,897,
    495,334,688,113,555,840,672,282,726,
    365,918,211,048,674,536,374,096,577
    ------------------------------------------------------------------------------
    dap_k(62): 1, ( 109
    digits)
    000,000,000,000,000,000,000,000,000,
    000,000,000,000,000,000,000,000,000,
    000,000,000,000,000,000,000,000,000,
    000,000,000,000,000,000,000,000,000
    ------------------------------------------------------------------------------
    dividend(63): 4,130,870,200,525,634, ( 43
    digits)
    701,470,856,139,171,214,138,369,936
    ------------------------------------------------------------------------------
    divisor(63): 2,372,114,765,269,637, ( 43
    digits)
    112,227,372,874,198,197,156,747,951
    ------------------------------------------------------------------------------
    a(63): 1 ( 1
    digit )
    ------------------------------------------------------------------------------
    p(63): 126,681, ( 33
    digits)
    625,842,748,498,335,889,192,663,895
    ------------------------------------------------------------------------------
    q(63): 397,982, ( 33
    digits)
    065,092,389,582,058,319,745,527,393
    ------------------------------------------------------------------------------
    dap_h(63): 318,309,886,183,790,671,537,767,526, ( 108
    digits)
    745,028,724,068,919,291,480,912,897,
    495,334,688,120,606,934,318,270,363,
    018,565,103,859,798,261,239,211,730
    ------------------------------------------------------------------------------
    dap_k(63): 1, ( 109
    digits)
    000,000,000,000,000,000,000,000,000,
    000,000,000,000,000,000,000,000,000,
    000,000,000,000,000,000,000,000,000,
    000,000,000,000,000,000,000,000,000
    ------------------------------------------------------------------------------
    dividend(64): 2,372,114,765,269,637, ( 43
    digits)
    112,227,372,874,198,197,156,747,951
    ------------------------------------------------------------------------------
    divisor(64): 1,758,755,435,255,997, ( 43
    digits)
    589,243,483,264,973,016,981,621,985
    ------------------------------------------------------------------------------
    a(64): 1 ( 1
    digit )
    ------------------------------------------------------------------------------
    p(64): 240,112, ( 33
    digits)
    203,426,728,897,943,499,903,682,238
    ------------------------------------------------------------------------------
    q(64): 754,334, ( 33
    digits)
    734,322,669,483,655,537,561,140,633
    ------------------------------------------------------------------------------
    dap_h(64): 318,309,886,183,790,671,537,767,526, ( 108
    digits)
    745,028,724,068,919,291,480,912,897,
    495,334,688,117,275,951,496,703,042,
    606,638,859,394,599,991,855,822,150
    ------------------------------------------------------------------------------
    dap_k(64): 1, ( 109
    digits)
    000,000,000,000,000,000,000,000,000,
    000,000,000,000,000,000,000,000,000,
    000,000,000,000,000,000,000,000,000,
    000,000,000,000,000,000,000,000,000
    ------------------------------------------------------------------------------
    dividend(65): 1,758,755,435,255,997, ( 43
    digits)
    589,243,483,264,973,016,981,621,985
    ------------------------------------------------------------------------------
    divisor(65): 613,359,330,013,639, ( 42
    digits)
    522,983,889,609,225,180,175,125,966
    ------------------------------------------------------------------------------
    a(65): 2 ( 1
    digit )
    ------------------------------------------------------------------------------
    p(65): 606,906, ( 33
    digits)
    032,696,206,294,222,889,000,028,371
    ------------------------------------------------------------------------------
    q(65): 1,906,651, ( 34
    digits)
    533,737,728,549,369,394,867,808,659
    ------------------------------------------------------------------------------
    dap_h(65): 318,309,886,183,790,671,537,767,526, ( 108
    digits)
    745,028,724,068,919,291,480,912,897,
    495,334,688,117,971,239,229,493,858,
    354,655,778,669,658,822,433,207,181
    ------------------------------------------------------------------------------
    dap_k(65): 1, ( 109
    digits)
    000,000,000,000,000,000,000,000,000,
    000,000,000,000,000,000,000,000,000,
    000,000,000,000,000,000,000,000,000,
    000,000,000,000,000,000,000,000,000
    ------------------------------------------------------------------------------
    dividend(66): 613,359,330,013,639, ( 42
    digits)
    522,983,889,609,225,180,175,125,966
    ------------------------------------------------------------------------------
    divisor(66): 532,036,775,228,718, ( 42
    digits)
    543,275,704,046,522,656,631,370,053
    ------------------------------------------------------------------------------
    a(66): 1 ( 1
    digit )
    ------------------------------------------------------------------------------
    p(66): 847,018, ( 33
    digits)
    236,122,935,192,166,388,903,710,609
    ------------------------------------------------------------------------------
    q(66): 2,660,986, ( 34
    digits)
    268,060,398,033,024,932,428,949,292
    ------------------------------------------------------------------------------
    dap_h(66): 318,309,886,183,790,671,537,767,526, ( 108
    digits)
    745,028,724,068,919,291,480,912,897,
    495,334,688,117,774,139,494,900,013,
    102,848,642,611,257,968,324,202,408
    ------------------------------------------------------------------------------
    dap_k(66): 1, ( 109
    digits)
    000,000,000,000,000,000,000,000,000,
    000,000,000,000,000,000,000,000,000,
    000,000,000,000,000,000,000,000,000,
    000,000,000,000,000,000,000,000,000
    ------------------------------------------------------------------------------
    dividend(67): 532,036,775,228,718, ( 42
    digits)
    543,275,704,046,522,656,631,370,053
    ------------------------------------------------------------------------------
    divisor(67): 81,322,554,784,920, ( 41
    digits)
    979,708,185,562,702,523,543,755,913
    ------------------------------------------------------------------------------
    a(67): 6 ( 1
    digit )
    ------------------------------------------------------------------------------
    p(67): 5,689,015, ( 34
    digits)
    449,433,817,447,221,222,422,292,025
    ------------------------------------------------------------------------------
    q(67): 17,872,569, ( 35
    digits)
    142,100,116,747,518,989,441,504,411
    ------------------------------------------------------------------------------
    dap_h(67): 318,309,886,183,790,671,537,767,526, ( 108
    digits)
    745,028,724,068,919,291,480,912,897,
    495,334,688,117,795,166,159,185,083,
    863,145,408,802,752,915,780,858,169
    ------------------------------------------------------------------------------
    dap_k(67): 1, ( 109
    digits)
    000,000,000,000,000,000,000,000,000,
    000,000,000,000,000,000,000,000,000,
    000,000,000,000,000,000,000,000,000,
    000,000,000,000,000,000,000,000,000
    ------------------------------------------------------------------------------
    dividend(68): 81,322,554,784,920, ( 41
    digits)
    979,708,185,562,702,523,543,755,913
    ------------------------------------------------------------------------------
    divisor(68): 44,101,446,519,192, ( 41
    digits)
    665,026,590,670,307,515,368,834,575
    ------------------------------------------------------------------------------
    a(68): 1 ( 1
    digit )
    ------------------------------------------------------------------------------
    p(68): 6,536,033, ( 34
    digits)
    685,556,752,639,387,611,326,002,634
    ------------------------------------------------------------------------------
    q(68): 20,533,555, ( 35
    digits)
    410,160,514,780,543,921,870,453,703
    ------------------------------------------------------------------------------
    dap_h(68): 318,309,886,183,790,671,537,767,526, ( 108
    digits)
    745,028,724,068,919,291,480,912,897,
    495,334,688,117,792,441,269,909,551,
    822,456,414,465,575,430,876,286,571
    ------------------------------------------------------------------------------
    dap_k(68): 1, ( 109
    digits)
    000,000,000,000,000,000,000,000,000,
    000,000,000,000,000,000,000,000,000,
    000,000,000,000,000,000,000,000,000,
    000,000,000,000,000,000,000,000,000
    ------------------------------------------------------------------------------
    dividend(69): 44,101,446,519,192, ( 41
    digits)
    665,026,590,670,307,515,368,834,575
    ------------------------------------------------------------------------------
    divisor(69): 37,221,108,265,728, ( 41
    digits)
    314,681,594,892,395,008,174,921,338
    ------------------------------------------------------------------------------
    a(69): 1 ( 1
    digit )
    ------------------------------------------------------------------------------
    p(69): 12,225,049, ( 35
    digits)
    134,990,570,086,608,833,748,294,659
    ------------------------------------------------------------------------------
    q(69): 38,406,124, ( 35
    digits)
    552,260,631,528,062,911,311,958,114
    ------------------------------------------------------------------------------
    dap_h(69): 318,309,886,183,790,671,537,767,526, ( 108
    digits)
    745,028,724,068,919,291,480,912,897,
    495,334,688,117,793,709,316,934,584,
    107,470,398,514,892,896,310,064,758
    ------------------------------------------------------------------------------
    dap_k(69): 1, ( 109
    digits)
    000,000,000,000,000,000,000,000,000,
    000,000,000,000,000,000,000,000,000,
    000,000,000,000,000,000,000,000,000,
    000,000,000,000,000,000,000,000,000
    ------------------------------------------------------------------------------
    dividend(70): 37,221,108,265,728, ( 41
    digits)
    314,681,594,892,395,008,174,921,338
    ------------------------------------------------------------------------------
    divisor(70): 6,880,338,253,464, ( 40
    digits)
    350,344,995,777,912,507,193,913,237
    ------------------------------------------------------------------------------
    a(70): 5 ( 1
    digit )
    ------------------------------------------------------------------------------
    p(70): 67,661,279, ( 35
    digits)
    360,509,603,072,431,780,067,475,929
    ------------------------------------------------------------------------------
    q(70): 212,564,178, ( 36
    digits)
    171,463,672,420,858,478,430,244,273
    ------------------------------------------------------------------------------
    dap_h(70): 318,309,886,183,790,671,537,767,526, ( 108
    digits)
    745,028,724,068,919,291,480,912,897,
    495,334,688,117,793,586,824,452,195,
    157,575,044,369,893,407,494,325,471
    ------------------------------------------------------------------------------
    dap_k(70): 1, ( 109
    digits)
    000,000,000,000,000,000,000,000,000,
    000,000,000,000,000,000,000,000,000,
    000,000,000,000,000,000,000,000,000,
    000,000,000,000,000,000,000,000,000
    ------------------------------------------------------------------------------
    dividend(71): 6,880,338,253,464, ( 40
    digits)
    350,344,995,777,912,507,193,913,237
    ------------------------------------------------------------------------------
    divisor(71): 2,819,416,998,406, ( 40
    digits)
    562,956,616,002,832,472,205,355,153
    ------------------------------------------------------------------------------
    a(71): 2 ( 1
    digit )
    ------------------------------------------------------------------------------
    p(71): 147,547,607, ( 36
    digits)
    856,009,776,231,472,393,883,246,517
    ------------------------------------------------------------------------------
    q(71): 463,534,480, ( 36
    digits)
    895,187,976,369,779,868,172,446,660
    ------------------------------------------------------------------------------
    dap_h(71): 318,309,886,183,790,671,537,767,526, ( 108
    digits)
    745,028,724,068,919,291,480,912,897,
    495,334,688,117,793,596,973,560,253,
    441,053,951,196,623,767,333,415,071
    ------------------------------------------------------------------------------
    dap_k(71): 1, ( 109
    digits)
    000,000,000,000,000,000,000,000,000,
    000,000,000,000,000,000,000,000,000,
    000,000,000,000,000,000,000,000,000,
    000,000,000,000,000,000,000,000,000
    ------------------------------------------------------------------------------
    dividend(72): 2,819,416,998,406, ( 40
    digits)
    562,956,616,002,832,472,205,355,153
    ------------------------------------------------------------------------------
    divisor(72): 1,241,504,256,651, ( 40
    digits)
    224,431,763,772,247,562,783,202,931
    ------------------------------------------------------------------------------
    a(72): 2 ( 1
    digit )
    ------------------------------------------------------------------------------
    p(72): 362,756,495, ( 36
    digits)
    072,529,155,535,376,567,833,968,963
    ------------------------------------------------------------------------------
    q(72): 1,139,633,139, ( 37
    digits)
    961,839,625,160,418,214,775,137,593
    ------------------------------------------------------------------------------
    dap_h(72): 318,309,886,183,790,671,537,767,526, ( 108
    digits)
    745,028,724,068,919,291,480,912,897,
    495,334,688,117,793,595,080,550,355,
    839,580,996,885,303,413,255,964,557
    ------------------------------------------------------------------------------
    dap_k(72): 1, ( 109
    digits)
    000,000,000,000,000,000,000,000,000,
    000,000,000,000,000,000,000,000,000,
    000,000,000,000,000,000,000,000,000,
    000,000,000,000,000,000,000,000,000
    ------------------------------------------------------------------------------
    dividend(73): 1,241,504,256,651, ( 40
    digits)
    224,431,763,772,247,562,783,202,931
    ------------------------------------------------------------------------------
    divisor(73): 336,408,485,104, ( 39
    digits)
    114,093,088,458,337,346,638,949,291
    ------------------------------------------------------------------------------
    a(73): 3 ( 1
    digit )
    ------------------------------------------------------------------------------
    p(73): 1,235,817,093, ( 37
    digits)
    073,597,242,837,602,097,385,153,406
    ------------------------------------------------------------------------------
    q(73): 3,882,433,900, ( 37
    digits)
    780,706,851,851,034,512,497,859,439
    ------------------------------------------------------------------------------
    dap_h(73): 318,309,886,183,790,671,537,767,526, ( 108
    digits)
    745,028,724,068,919,291,480,912,897,
    495,334,688,117,793,595,306,562,023,
    427,199,901,384,529,006,394,765,969
    ------------------------------------------------------------------------------
    dap_k(73): 1, ( 109
    digits)
    000,000,000,000,000,000,000,000,000,
    000,000,000,000,000,000,000,000,000,
    000,000,000,000,000,000,000,000,000,
    000,000,000,000,000,000,000,000,000
    ------------------------------------------------------------------------------
    dividend(74): 336,408,485,104, ( 39
    digits)
    114,093,088,458,337,346,638,949,291
    ------------------------------------------------------------------------------
    divisor(74): 232,278,801,338, ( 39
    digits)
    882,152,498,397,235,522,866,355,058
    ------------------------------------------------------------------------------
    a(74): 1 ( 1
    digit )
    ------------------------------------------------------------------------------
    p(74): 1,598,573,588, ( 37
    digits)
    146,126,398,372,978,665,219,122,369
    ------------------------------------------------------------------------------
    q(74): 5,022,067,040, ( 37
    digits)
    742,546,477,011,452,727,272,997,032
    ------------------------------------------------------------------------------
    dap_h(74): 318,309,886,183,790,671,537,767,526, ( 108
    digits)
    745,028,724,068,919,291,480,912,897,
    495,334,688,117,793,595,255,274,299,
    804,377,425,320,074,437,694,319,045
    ------------------------------------------------------------------------------
    dap_k(74): 1, ( 109
    digits)
    000,000,000,000,000,000,000,000,000,
    000,000,000,000,000,000,000,000,000,
    000,000,000,000,000,000,000,000,000,
    000,000,000,000,000,000,000,000,000
    ------------------------------------------------------------------------------
    dividend(75): 232,278,801,338, ( 39
    digits)
    882,152,498,397,235,522,866,355,058
    ------------------------------------------------------------------------------
    divisor(75): 104,129,683,765, ( 39
    digits)
    231,940,590,061,101,823,772,594,233
    ------------------------------------------------------------------------------
    a(75): 2 ( 1
    digit )
    ------------------------------------------------------------------------------
    p(75): 4,432,964,269, ( 37
    digits)
    365,850,039,583,559,427,823,398,144
    ------------------------------------------------------------------------------
    q(75): 13,926,567,982, ( 38
    digits)
    265,799,805,873,939,967,043,853,503
    ------------------------------------------------------------------------------
    dap_h(75): 318,309,886,183,790,671,537,767,526, ( 108
    digits)
    745,028,724,068,919,291,480,912,897,
    495,334,688,117,793,595,269,572,237,
    181,388,039,551,070,944,293,476,802
    ------------------------------------------------------------------------------
    dap_k(75): 1, ( 109
    digits)
    000,000,000,000,000,000,000,000,000,
    000,000,000,000,000,000,000,000,000,
    000,000,000,000,000,000,000,000,000,
    000,000,000,000,000,000,000,000,000
    ------------------------------------------------------------------------------
    dividend(76): 104,129,683,765, ( 39
    digits)
    231,940,590,061,101,823,772,594,233
    ------------------------------------------------------------------------------
    divisor(76): 24,019,433,808, ( 38
    digits)
    418,271,318,275,031,875,321,166,592
    ------------------------------------------------------------------------------
    a(76): 4 ( 1
    digit )
    ------------------------------------------------------------------------------
    p(76): 19,330,430,665, ( 38
    digits)
    609,526,556,707,216,376,512,714,945
    ------------------------------------------------------------------------------
    q(76): 60,728,338,969, ( 38
    digits)
    805,745,700,507,212,595,448,411,044
    ------------------------------------------------------------------------------
    dap_h(76): 318,309,886,183,790,671,537,767,526, ( 108
    digits)
    745,028,724,068,919,291,480,912,897,
    495,334,688,117,793,595,268,389,836,
    982,901,009,718,683,132,226,852,995
    ------------------------------------------------------------------------------
    dap_k(76): 1, ( 109
    digits)
    000,000,000,000,000,000,000,000,000,
    000,000,000,000,000,000,000,000,000,
    000,000,000,000,000,000,000,000,000,
    000,000,000,000,000,000,000,000,000
    ------------------------------------------------------------------------------
    dividend(77): 24,019,433,808, ( 38
    digits)
    418,271,318,275,031,875,321,166,592
    ------------------------------------------------------------------------------
    divisor(77): 8,051,948,531, ( 37
    digits)
    558,855,316,960,974,322,487,927,865
    ------------------------------------------------------------------------------
    a(77): 2 ( 1
    digit )
    ------------------------------------------------------------------------------
    p(77): 43,093,825,600, ( 38
    digits)
    584,903,152,997,992,180,848,828,034
    ------------------------------------------------------------------------------
    q(77): 135,383,245,921, ( 39
    digits)
    877,291,206,888,365,157,940,675,591
    ------------------------------------------------------------------------------
    dap_h(77): 318,309,886,183,790,671,537,767,526, ( 108
    digits)
    745,028,724,068,919,291,480,912,897,
    495,334,688,117,793,595,268,511,467,
    814,207,686,515,778,613,953,474,880
    ------------------------------------------------------------------------------
    dap_k(77): 1, ( 109
    digits)
    000,000,000,000,000,000,000,000,000,
    000,000,000,000,000,000,000,000,000,
    000,000,000,000,000,000,000,000,000,
    000,000,000,000,000,000,000,000,000
    ------------------------------------------------------------------------------
    dividend(78): 8,051,948,531, ( 37
    digits)
    558,855,316,960,974,322,487,927,865
    ------------------------------------------------------------------------------
    divisor(78): 7,915,536,745, ( 37
    digits)
    300,560,684,353,083,230,345,310,862
    ------------------------------------------------------------------------------
    a(78): 1 ( 1
    digit )
    ------------------------------------------------------------------------------
    p(78): 62,424,256,266, ( 38
    digits)
    194,429,709,705,208,557,361,542,979
    ------------------------------------------------------------------------------
    q(78): 196,111,584,891, ( 39
    digits)
    683,036,907,395,577,753,389,086,635
    ------------------------------------------------------------------------------
    dap_h(78): 318,309,886,183,790,671,537,767,526, ( 108
    digits)
    745,028,724,068,919,291,480,912,897,
    495,334,688,117,793,595,268,473,803,
    347,027,683,713,048,958,303,614,025
    ------------------------------------------------------------------------------
    dap_k(78): 1, ( 109
    digits)
    000,000,000,000,000,000,000,000,000,
    000,000,000,000,000,000,000,000,000,
    000,000,000,000,000,000,000,000,000,
    000,000,000,000,000,000,000,000,000
    ------------------------------------------------------------------------------
    dividend(79): 7,915,536,745, ( 37
    digits)
    300,560,684,353,083,230,345,310,862
    ------------------------------------------------------------------------------
    divisor(79): 136,411,786, ( 36
    digits)
    258,294,632,607,891,092,142,617,003
    ------------------------------------------------------------------------------
    a(79): 58 ( 2
    digits)
    ------------------------------------------------------------------------------
    p(79): 3,663,700,689,039, ( 40
    digits)
    861,826,315,900,088,507,818,320,816
    ------------------------------------------------------------------------------
    q(79): 11,509,855,169,639, ( 41
    digits)
    493,431,835,831,874,854,507,700,421
    ------------------------------------------------------------------------------
    dap_h(79): 318,309,886,183,790,671,537,767,526, ( 108
    digits)
    745,028,724,068,919,291,480,912,897,
    495,334,688,117,793,595,268,474,246,
    370,658,093,695,735,986,511,150,821
    ------------------------------------------------------------------------------
    dap_k(79): 1, ( 109
    digits)
    000,000,000,000,000,000,000,000,000,
    000,000,000,000,000,000,000,000,000,
    000,000,000,000,000,000,000,000,000,
    000,000,000,000,000,000,000,000,000
    ------------------------------------------------------------------------------
    dividend(80): 136,411,786, ( 36
    digits)
    258,294,632,607,891,092,142,617,003
    ------------------------------------------------------------------------------
    divisor(80): 3,653,142, ( 34
    digits)
    319,471,993,095,399,886,073,524,688
    ------------------------------------------------------------------------------
    a(80): 37 ( 2
    digits)
    ------------------------------------------------------------------------------
    p(80): 135,619,349,750,741, ( 42
    digits)
    082,003,398,008,483,346,639,413,171
    ------------------------------------------------------------------------------
    q(80): 426,060,752,861,552, ( 42
    digits)
    940,014,833,174,947,370,174,002,212
    ------------------------------------------------------------------------------
    dap_h(80): 318,309,886,183,790,671,537,767,526, ( 108
    digits)
    745,028,724,068,919,291,480,912,897,
    495,334,688,117,793,595,268,474,246,
    166,738,662,674,120,681,685,874,728
    ------------------------------------------------------------------------------
    dap_k(80): 1, ( 109
    digits)
    000,000,000,000,000,000,000,000,000,
    000,000,000,000,000,000,000,000,000,
    000,000,000,000,000,000,000,000,000,
    000,000,000,000,000,000,000,000,000
    ------------------------------------------------------------------------------
    dividend(81): 3,653,142, ( 34
    digits)
    319,471,993,095,399,886,073,524,688
    ------------------------------------------------------------------------------
    divisor(81): 1,245,520, ( 34
    digits)
    437,830,888,078,095,307,422,203,547
    ------------------------------------------------------------------------------
    a(81): 2 ( 1
    digit )
    ------------------------------------------------------------------------------
    p(81): 274,902,400,190,522, ( 42
    digits)
    025,833,111,917,055,201,097,147,158
    ------------------------------------------------------------------------------
    q(81): 863,631,360,892,745, ( 42
    digits)
    373,461,502,181,769,594,855,704,845
    ------------------------------------------------------------------------------
    dap_h(81): 318,309,886,183,790,671,537,767,526, ( 108
    digits)
    745,028,724,068,919,291,480,912,897,
    495,334,688,117,793,595,268,474,246,
    169,456,353,605,246,523,385,826,268
    ------------------------------------------------------------------------------
    dap_k(81): 1, ( 109
    digits)
    000,000,000,000,000,000,000,000,000,
    000,000,000,000,000,000,000,000,000,
    000,000,000,000,000,000,000,000,000,
    000,000,000,000,000,000,000,000,000
    ------------------------------------------------------------------------------
    dividend(82): 1,245,520, ( 34
    digits)
    437,830,888,078,095,307,422,203,547
    ------------------------------------------------------------------------------
    divisor(82): 1,162,101, ( 34
    digits)
    443,810,216,939,209,271,229,117,594
    ------------------------------------------------------------------------------
    a(82): 1 ( 1
    digit )
    ------------------------------------------------------------------------------
    p(82): 410,521,749,941,263, ( 42
    digits)
    107,836,509,925,538,547,736,560,329
    ------------------------------------------------------------------------------
    q(82): 1,289,692,113,754,298, ( 43
    digits)
    313,476,335,356,716,965,029,707,057
    ------------------------------------------------------------------------------
    dap_h(82): 318,309,886,183,790,671,537,767,526, ( 108
    digits)
    745,028,724,068,919,291,480,912,897,
    495,334,688,117,793,595,268,474,246,
    168,558,541,304,301,427,485,517,978
    ------------------------------------------------------------------------------
    dap_k(82): 1, ( 109
    digits)
    000,000,000,000,000,000,000,000,000,
    000,000,000,000,000,000,000,000,000,
    000,000,000,000,000,000,000,000,000,
    000,000,000,000,000,000,000,000,000
    ------------------------------------------------------------------------------
    dividend(83): 1,162,101, ( 34
    digits)
    443,810,216,939,209,271,229,117,594
    ------------------------------------------------------------------------------
    divisor(83): 83,418, ( 32
    digits)
    994,020,671,138,886,036,193,085,953
    ------------------------------------------------------------------------------
    a(83): 13 ( 2
    digits)
    ------------------------------------------------------------------------------
    p(83): 5,611,685,149,426,942, ( 43
    digits)
    427,707,740,949,056,321,672,431,435
    ------------------------------------------------------------------------------
    q(83): 17,629,628,839,698,623, ( 44
    digits)
    448,653,861,819,090,140,241,896,586
    ------------------------------------------------------------------------------
    dap_h(83): 318,309,886,183,790,671,537,767,526, ( 108
    digits)
    745,028,724,068,919,291,480,912,897,
    495,334,688,117,793,595,268,474,246,
    168,602,522,880,223,516,793,173,158
    ------------------------------------------------------------------------------
    dap_k(83): 1, ( 109
    digits)
    000,000,000,000,000,000,000,000,000,
    000,000,000,000,000,000,000,000,000,
    000,000,000,000,000,000,000,000,000,
    000,000,000,000,000,000,000,000,000
    ------------------------------------------------------------------------------
    dividend(84): 83,418, ( 32
    digits)
    994,020,671,138,886,036,193,085,953
    ------------------------------------------------------------------------------
    divisor(84): 77,654, ( 32
    digits)
    521,541,492,133,690,800,719,000,205
    ------------------------------------------------------------------------------
    a(84): 1 ( 1
    digit )
    ------------------------------------------------------------------------------
    p(84): 6,022,206,899,368,205, ( 43
    digits)
    535,544,250,874,594,869,408,991,764
    ------------------------------------------------------------------------------
    q(84): 18,919,320,953,452,921, ( 44
    digits)
    762,130,197,175,807,105,271,603,643
    ------------------------------------------------------------------------------
    dap_h(84): 318,309,886,183,790,671,537,767,526, ( 108
    digits)
    745,028,724,068,919,291,480,912,897,
    495,334,688,117,793,595,268,474,246,
    168,599,524,744,521,037,884,536,975
    ------------------------------------------------------------------------------
    dap_k(84): 1, ( 109
    digits)
    000,000,000,000,000,000,000,000,000,
    000,000,000,000,000,000,000,000,000,
    000,000,000,000,000,000,000,000,000,
    000,000,000,000,000,000,000,000,000
    ------------------------------------------------------------------------------
    dividend(85): 77,654, ( 32
    digits)
    521,541,492,133,690,800,719,000,205
    ------------------------------------------------------------------------------
    divisor(85): 5,764, ( 31
    digits)
    472,479,179,005,195,235,474,085,748
    ------------------------------------------------------------------------------
    a(85): 13 ( 2
    digits)
    ------------------------------------------------------------------------------
    p(85): 83,900,374,841,213,614, ( 44
    digits)
    389,783,002,318,789,623,989,324,367
    ------------------------------------------------------------------------------
    q(85): 263,580,801,234,586,606, ( 45
    digits)
    356,346,425,104,582,508,772,743,945
    ------------------------------------------------------------------------------
    dap_h(85): 318,309,886,183,790,671,537,767,526, ( 108
    digits)
    745,028,724,068,919,291,480,912,897,
    495,334,688,117,793,595,268,474,246,
    168,599,725,275,134,034,657,634,315
    ------------------------------------------------------------------------------
    dap_k(85): 1, ( 109
    digits)
    000,000,000,000,000,000,000,000,000,
    000,000,000,000,000,000,000,000,000,
    000,000,000,000,000,000,000,000,000,
    000,000,000,000,000,000,000,000,000
    ------------------------------------------------------------------------------
    dividend(86): 5,764, ( 31
    digits)
    472,479,179,005,195,235,474,085,748
    ------------------------------------------------------------------------------
    divisor(86): 2,716, ( 31
    digits)
    379,312,165,066,152,739,555,885,481
    ------------------------------------------------------------------------------
    a(86): 2 ( 1
    digit )
    ------------------------------------------------------------------------------
    p(86): 173,822,956,581,795,434, ( 45
    digits)
    315,110,255,512,174,117,387,640,498
    ------------------------------------------------------------------------------
    q(86): 546,080,923,422,626,134, ( 45
    digits)
    474,823,047,384,972,122,817,091,533
    ------------------------------------------------------------------------------
    dap_h(86): 318,309,886,183,790,671,537,767,526, ( 108
    digits)
    745,028,724,068,919,291,480,912,897,
    495,334,688,117,793,595,268,474,246,
    168,599,718,327,623,390,081,595,001
    ------------------------------------------------------------------------------
    dap_k(86): 1, ( 109
    digits)
    000,000,000,000,000,000,000,000,000,
    000,000,000,000,000,000,000,000,000,
    000,000,000,000,000,000,000,000,000,
    000,000,000,000,000,000,000,000,000
    ------------------------------------------------------------------------------
    dividend(87): 2,716, ( 31
    digits)
    379,312,165,066,152,739,555,885,481
    ------------------------------------------------------------------------------
    divisor(87): 331, ( 30
    digits)
    713,854,848,872,889,756,362,314,786
    ------------------------------------------------------------------------------
    a(87): 8 ( 1
    digit )
    ------------------------------------------------------------------------------
    p(87): 1,474,484,027,495,577,088, ( 46
    digits)
    910,665,046,416,182,563,090,448,351
    ------------------------------------------------------------------------------
    q(87): 4,632,228,188,615,595,682, ( 46
    digits)
    154,930,804,184,359,491,309,476,209
    ------------------------------------------------------------------------------
    dap_h(87): 318,309,886,183,790,671,537,767,526, ( 108
    digits)
    745,028,724,068,919,291,480,912,897,
    495,334,688,117,793,595,268,474,246,
    168,599,718,722,947,270,451,672,318
    ------------------------------------------------------------------------------
    dap_k(87): 1, ( 109
    digits)
    000,000,000,000,000,000,000,000,000,
    000,000,000,000,000,000,000,000,000,
    000,000,000,000,000,000,000,000,000,
    000,000,000,000,000,000,000,000,000
    ------------------------------------------------------------------------------
    dividend(88): 331, ( 30
    digits)
    713,854,848,872,889,756,362,314,786
    ------------------------------------------------------------------------------
    divisor(88): 62, ( 29
    digits)
    668,473,374,083,034,688,657,367,193
    ------------------------------------------------------------------------------
    a(88): 5 ( 1
    digit )
    ------------------------------------------------------------------------------
    p(88): 7,546,243,094,059,680,878, ( 46
    digits)
    868,435,487,593,086,932,839,882,253
    ------------------------------------------------------------------------------
    q(88): 23,707,221,866,500,604,545, ( 47
    digits)
    249,477,068,306,769,579,364,472,578
    ------------------------------------------------------------------------------
    dap_h(88): 318,309,886,183,790,671,537,767,526, ( 108
    digits)
    745,028,724,068,919,291,480,912,897,
    495,334,688,117,793,595,268,474,246,
    168,599,718,713,841,233,866,738,073
    ------------------------------------------------------------------------------
    dap_k(88): 1, ( 109
    digits)
    000,000,000,000,000,000,000,000,000,
    000,000,000,000,000,000,000,000,000,
    000,000,000,000,000,000,000,000,000,
    000,000,000,000,000,000,000,000,000
    ------------------------------------------------------------------------------
    dividend(89): 62, ( 29
    digits)
    668,473,374,083,034,688,657,367,193
    ------------------------------------------------------------------------------
    divisor(89): 18, ( 29
    digits)
    371,487,978,457,716,313,075,478,821
    ------------------------------------------------------------------------------
    a(89): 3 ( 1
    digit )
    ------------------------------------------------------------------------------
    p(89): 24,113,213,309,674,619,725, ( 47
    digits)
    515,971,509,195,443,361,610,095,110
    ------------------------------------------------------------------------------
    q(89): 75,753,893,788,117,409,317, ( 47
    digits)
    903,362,009,104,668,229,402,893,943
    ------------------------------------------------------------------------------
    dap_h(89): 318,309,886,183,790,671,537,767,526, ( 108
    digits)
    745,028,724,068,919,291,480,912,897,
    495,334,688,117,793,595,268,474,246,
    168,599,718,714,398,053,288,763,793
    ------------------------------------------------------------------------------
    dap_k(89): 1, ( 109
    digits)
    000,000,000,000,000,000,000,000,000,
    000,000,000,000,000,000,000,000,000,
    000,000,000,000,000,000,000,000,000,
    000,000,000,000,000,000,000,000,000
    ------------------------------------------------------------------------------
    dividend(90): 18, ( 29
    digits)
    371,487,978,457,716,313,075,478,821
    ------------------------------------------------------------------------------
    divisor(90): 7, ( 28
    digits)
    554,009,438,709,885,749,430,930,730
    ------------------------------------------------------------------------------
    a(90): 2 ( 1
    digit )
    ------------------------------------------------------------------------------
    p(90): 55,772,669,713,408,920,329, ( 47
    digits)
    900,378,505,983,973,656,060,072,473
    ------------------------------------------------------------------------------
    q(90): 175,215,009,442,735,423,181, ( 48
    digits)
    056,201,086,516,106,038,170,260,464
    ------------------------------------------------------------------------------
    dap_h(90): 318,309,886,183,790,671,537,767,526, ( 108
    digits)
    745,028,724,068,919,291,480,912,897,
    495,334,688,117,793,595,268,474,246,
    168,599,718,714,322,713,615,413,180
    ------------------------------------------------------------------------------
    dap_k(90): 1, ( 109
    digits)
    000,000,000,000,000,000,000,000,000,
    000,000,000,000,000,000,000,000,000,
    000,000,000,000,000,000,000,000,000,
    000,000,000,000,000,000,000,000,000
    ------------------------------------------------------------------------------
    dividend(91): 7, ( 28
    digits)
    554,009,438,709,885,749,430,930,730
    ------------------------------------------------------------------------------
    divisor(91): 3, ( 28
    digits)
    263,469,101,037,944,814,213,617,361
    ------------------------------------------------------------------------------
    a(91): 2 ( 1
    digit )
    ------------------------------------------------------------------------------
    p(91): 135,658,552,736,492,460,385, ( 48
    digits)
    316,728,521,163,390,673,730,240,056
    ------------------------------------------------------------------------------
    q(91): 426,183,912,673,588,255,680, ( 48
    digits)
    015,764,182,136,880,305,743,414,871
    ------------------------------------------------------------------------------
    dap_h(91): 318,309,886,183,790,671,537,767,526, ( 108
    digits)
    745,028,724,068,919,291,480,912,897,
    495,334,688,117,793,595,268,474,246,
    168,599,718,714,336,105,189,903,240
    ------------------------------------------------------------------------------
    dap_k(91): 1, ( 109
    digits)
    000,000,000,000,000,000,000,000,000,
    000,000,000,000,000,000,000,000,000,
    000,000,000,000,000,000,000,000,000,
    000,000,000,000,000,000,000,000,000
    ------------------------------------------------------------------------------
    dividend(92): 3, ( 28
    digits)
    263,469,101,037,944,814,213,617,361
    ------------------------------------------------------------------------------
    divisor(92): 1, ( 28
    digits)
    027,071,236,633,996,121,003,696,008
    ------------------------------------------------------------------------------
    a(92): 3 ( 1
    digit )
    ------------------------------------------------------------------------------
    p(92): 462,748,327,922,886,301,485, ( 48
    digits)
    850,564,069,474,145,677,250,792,641
    ------------------------------------------------------------------------------
    q(92): 1,453,766,747,463,500,190,221, ( 49
    digits)
    103,493,632,926,746,955,400,505,077
    ------------------------------------------------------------------------------
    dap_h(92): 318,309,886,183,790,671,537,767,526, ( 108
    digits)
    745,028,724,068,919,291,480,912,897,
    495,334,688,117,793,595,268,474,246,
    168,599,718,714,334,491,172,486,194
    ------------------------------------------------------------------------------
    dap_k(92): 1, ( 109
    digits)
    000,000,000,000,000,000,000,000,000,
    000,000,000,000,000,000,000,000,000,
    000,000,000,000,000,000,000,000,000,
    000,000,000,000,000,000,000,000,000
    ------------------------------------------------------------------------------
    dividend(93): 1, ( 28
    digits)
    027,071,236,633,996,121,003,696,008
    ------------------------------------------------------------------------------
    divisor(93): 182,255,391,135,956,451,202,529,337 ( 27
    digits)
    ------------------------------------------------------------------------------
    a(93): 5 ( 1
    digit )
    ------------------------------------------------------------------------------
    p(93): 2,449,400,192,350,923,967,814, ( 49
    digits)
    569,548,868,534,119,059,984,203,261
    ------------------------------------------------------------------------------
    q(93): 7,695,017,649,991,089,206,785, ( 49
    digits)
    533,232,346,770,615,082,745,940,256
    ------------------------------------------------------------------------------
    dap_h(93): 318,309,886,183,790,671,537,767,526, ( 108
    digits)
    745,028,724,068,919,291,480,912,897,
    495,334,688,117,793,595,268,474,246,
    168,599,718,714,334,580,563,867,631
    ------------------------------------------------------------------------------
    dap_k(93): 1, ( 109
    digits)
    000,000,000,000,000,000,000,000,000,
    000,000,000,000,000,000,000,000,000,
    000,000,000,000,000,000,000,000,000,
    000,000,000,000,000,000,000,000,000
    ------------------------------------------------------------------------------
    dividend(94): 182,255,391,135,956,451,202,529,337 ( 27
    digits)
    ------------------------------------------------------------------------------
    divisor(94): 115,794,280,954,213,864,991,049,323 ( 27
    digits)
    ------------------------------------------------------------------------------
    a(94): 1 ( 1
    digit )
    ------------------------------------------------------------------------------
    p(94): 2,912,148,520,273,810,269,300, ( 49
    digits)
    420,112,938,008,264,737,234,995,902
    ------------------------------------------------------------------------------
    q(94): 9,148,784,397,454,589,397,006, ( 49
    digits)
    636,725,979,697,362,038,146,445,333
    ------------------------------------------------------------------------------
    dap_h(94): 318,309,886,183,790,671,537,767,526, ( 108
    digits)
    745,028,724,068,919,291,480,912,897,
    495,334,688,117,793,595,268,474,246,
    168,599,718,714,334,566,359,333,761
    ------------------------------------------------------------------------------
    dap_k(94): 1, ( 109
    digits)
    000,000,000,000,000,000,000,000,000,
    000,000,000,000,000,000,000,000,000,
    000,000,000,000,000,000,000,000,000,
    000,000,000,000,000,000,000,000,000
    ------------------------------------------------------------------------------
    dividend(95): 115,794,280,954,213,864,991,049,323 ( 27
    digits)
    ------------------------------------------------------------------------------
    divisor(95): 66,461,110,181,742,586,211,480,014 ( 26
    digits)
    ------------------------------------------------------------------------------
    a(95): 1 ( 1
    digit )
    ------------------------------------------------------------------------------
    p(95): 5,361,548,712,624,734,237,114, ( 49
    digits)
    989,661,806,542,383,797,219,199,163
    ------------------------------------------------------------------------------
    q(95): 16,843,802,047,445,678,603,792, ( 50
    digits)
    169,958,326,467,977,120,892,385,589
    ------------------------------------------------------------------------------
    dap_h(95): 318,309,886,183,790,671,537,767,526, ( 108
    digits)
    745,028,724,068,919,291,480,912,897,
    495,334,688,117,793,595,268,474,246,
    168,599,718,714,334,572,848,613,232
    ------------------------------------------------------------------------------
    dap_k(95): 1, ( 109
    digits)
    000,000,000,000,000,000,000,000,000,
    000,000,000,000,000,000,000,000,000,
    000,000,000,000,000,000,000,000,000,
    000,000,000,000,000,000,000,000,000
    ------------------------------------------------------------------------------
    dividend(96): 66,461,110,181,742,586,211,480,014 ( 26
    digits)
    ------------------------------------------------------------------------------
    divisor(96): 49,333,170,772,471,278,779,569,309 ( 26
    digits)
    ------------------------------------------------------------------------------
    a(96): 1 ( 1
    digit )
    ------------------------------------------------------------------------------
    p(96): 8,273,697,232,898,544,506,415, ( 49
    digits)
    409,774,744,550,648,534,454,195,065
    ------------------------------------------------------------------------------
    q(96): 25,992,586,444,900,268,000,798, ( 50
    digits)
    806,684,306,165,339,159,038,830,922
    ------------------------------------------------------------------------------
    dap_h(96): 318,309,886,183,790,671,537,767,526, ( 108
    digits)
    745,028,724,068,919,291,480,912,897,
    495,334,688,117,793,595,268,474,246,
    168,599,718,714,334,570,564,538,160
    ------------------------------------------------------------------------------
    dap_k(96): 1, ( 109
    digits)
    000,000,000,000,000,000,000,000,000,
    000,000,000,000,000,000,000,000,000,
    000,000,000,000,000,000,000,000,000,
    000,000,000,000,000,000,000,000,000
    ------------------------------------------------------------------------------
    dividend(97): 49,333,170,772,471,278,779,569,309 ( 26
    digits)
    ------------------------------------------------------------------------------
    divisor(97): 17,127,939,409,271,307,431,910,705 ( 26
    digits)
    ------------------------------------------------------------------------------
    a(97): 2 ( 1
    digit )
    ------------------------------------------------------------------------------
    p(97): 21,908,943,178,421,823,249,945, ( 50
    digits)
    809,211,295,643,680,866,127,589,293
    ------------------------------------------------------------------------------
    q(97): 68,828,974,937,246,214,605,389, ( 50
    digits)
    783,326,938,798,655,438,970,047,433
    ------------------------------------------------------------------------------
    dap_h(97): 318,309,886,183,790,671,537,767,526, ( 108
    digits)
    745,028,724,068,919,291,480,912,897,
    495,334,688,117,793,595,268,474,246,
    168,599,718,714,334,571,123,496,192
    ------------------------------------------------------------------------------
    dap_k(97): 1, ( 109
    digits)
    000,000,000,000,000,000,000,000,000,
    000,000,000,000,000,000,000,000,000,
    000,000,000,000,000,000,000,000,000,
    000,000,000,000,000,000,000,000,000
    ------------------------------------------------------------------------------
    dividend(98): 17,127,939,409,271,307,431,910,705 ( 26
    digits)
    ------------------------------------------------------------------------------
    divisor(98): 15,077,291,953,928,663,915,747,899 ( 26
    digits)
    ------------------------------------------------------------------------------
    a(98): 1 ( 1
    digit )
    ------------------------------------------------------------------------------
    p(98): 30,182,640,411,320,367,756,361, ( 50
    digits)
    218,986,040,194,329,400,581,784,358
    ------------------------------------------------------------------------------
    q(98): 94,821,561,382,146,482,606,188, ( 50
    digits)
    590,011,244,963,994,598,008,878,355
    ------------------------------------------------------------------------------
    dap_h(98): 318,309,886,183,790,671,537,767,526, ( 108
    digits)
    745,028,724,068,919,291,480,912,897,
    495,334,688,117,793,595,268,474,246,
    168,599,718,714,334,570,970,274,026
    ------------------------------------------------------------------------------
    dap_k(98): 1, ( 109
    digits)
    000,000,000,000,000,000,000,000,000,
    000,000,000,000,000,000,000,000,000,
    000,000,000,000,000,000,000,000,000,
    000,000,000,000,000,000,000,000,000
    ------------------------------------------------------------------------------
    dividend(99): 15,077,291,953,928,663,915,747,899 ( 26
    digits)
    ------------------------------------------------------------------------------
    divisor(99): 2,050,647,455,342,643,516,162,806 ( 25
    digits)
    ------------------------------------------------------------------------------
    a(99): 7 ( 1
    digit )
    ------------------------------------------------------------------------------
    p(99): 233,187,426,057,664,397,544,474, ( 51
    digits)
    342,113,577,003,986,670,200,079,799
    ------------------------------------------------------------------------------
    q(99): 732,579,904,612,271,592,848,709, ( 51
    digits)
    913,405,653,546,617,625,032,195,918
    ------------------------------------------------------------------------------
    dap_h(99): 318,309,886,183,790,671,537,767,526, ( 108
    digits)
    745,028,724,068,919,291,480,912,897,
    495,334,688,117,793,595,268,474,246,
    168,599,718,714,334,570,984,669,896
    ------------------------------------------------------------------------------
    dap_k(99): 1, ( 109
    digits)
    000,000,000,000,000,000,000,000,000,
    000,000,000,000,000,000,000,000,000,
    000,000,000,000,000,000,000,000,000,
    000,000,000,000,000,000,000,000,000
    ------------------------------------------------------------------------------
    dividend(100): 2,050,647,455,342,643,516,162,806 ( 25
    digits)
    ------------------------------------------------------------------------------
    divisor(100): 722,759,766,530,159,302,608,257 ( 24
    digits)
    ------------------------------------------------------------------------------
    a(100): 2 ( 1
    digit )
    ------------------------------------------------------------------------------
    p(100): 496,557,492,526,649,162,845,309, ( 51
    digits)
    903,213,194,202,302,740,981,943,956
    ------------------------------------------------------------------------------
    q(100): 1,559,981,370,606,689,668,303,608, ( 52
    digits)
    416,822,552,057,229,848,073,270,191
    ------------------------------------------------------------------------------
    dap_h(100): 318,309,886,183,790,671,537,767,526, ( 108
    digits)
    745,028,724,068,919,291,480,912,897,
    495,334,688,117,793,595,268,474,246,
    168,599,718,714,334,570,983,794,860
    ------------------------------------------------------------------------------
    dap_k(100): 1, ( 109
    digits)
    000,000,000,000,000,000,000,000,000,
    000,000,000,000,000,000,000,000,000,
    000,000,000,000,000,000,000,000,000,
    000,000,000,000,000,000,000,000,000
    ------------------------------------------------------------------------------
    dividend(101): 722,759,766,530,159,302,608,257 ( 24
    digits)
    ------------------------------------------------------------------------------
    divisor(101): 605,127,922,282,324,910,946,292 ( 24
    digits)
    ------------------------------------------------------------------------------
    a(101): 1 ( 1
    digit )
    ------------------------------------------------------------------------------
    p(101): 729,744,918,584,313,560,389,784, ( 51
    digits)
    245,326,771,206,289,411,182,023,755
    ------------------------------------------------------------------------------
    q(101): 2,292,561,275,218,961,261,152,318, ( 52
    digits)
    330,228,205,603,847,473,105,466,109
    ------------------------------------------------------------------------------
    dap_h(101): 318,309,886,183,790,671,537,767,526, ( 108
    digits)
    745,028,724,068,919,291,480,912,897,
    495,334,688,117,793,595,268,474,246,
    168,599,718,714,334,570,984,074,475
    ------------------------------------------------------------------------------
    dap_k(101): 1, ( 109
    digits)
    000,000,000,000,000,000,000,000,000,
    000,000,000,000,000,000,000,000,000,
    000,000,000,000,000,000,000,000,000,
    000,000,000,000,000,000,000,000,000
    ------------------------------------------------------------------------------
    dividend(102): 605,127,922,282,324,910,946,292 ( 24
    digits)
    ------------------------------------------------------------------------------
    divisor(102): 117,631,844,247,834,391,661,965 ( 24
    digits)
    ------------------------------------------------------------------------------
    a(102): 5 ( 1
    digit )
    ------------------------------------------------------------------------------
    p(102): 4,145,282,085,448,216,964,794,231, ( 52
    digits)
    129,847,050,233,749,796,892,062,731
    ------------------------------------------------------------------------------
    q(102): 13,022,787,746,701,495,974,065,200, ( 53
    digits)
    067,963,580,076,467,213,600,600,736
    ------------------------------------------------------------------------------
    dap_h(102): 318,309,886,183,790,671,537,767,526, ( 108
    digits)
    745,028,724,068,919,291,480,912,897,
    495,334,688,117,793,595,268,474,246,
    168,599,718,714,334,570,984,040,980
    ------------------------------------------------------------------------------
    dap_k(102): 1, ( 109
    digits)
    000,000,000,000,000,000,000,000,000,
    000,000,000,000,000,000,000,000,000,
    000,000,000,000,000,000,000,000,000,
    000,000,000,000,000,000,000,000,000
    ------------------------------------------------------------------------------
    dividend(103): 117,631,844,247,834,391,661,965 ( 24
    digits)
    ------------------------------------------------------------------------------
    divisor(103): 16,968,701,043,152,952,636,467 ( 23
    digits)
    ------------------------------------------------------------------------------
    a(103): 6 ( 1
    digit )
    ------------------------------------------------------------------------------
    p(103): 25,601,437,431,273,615,349,155,171, ( 53
    digits)
    024,409,072,608,788,192,534,400,141
    ------------------------------------------------------------------------------
    q(103): 80,429,287,755,427,937,105,543,518, ( 53
    digits)
    738,009,686,062,650,754,709,070,525
    ------------------------------------------------------------------------------
    dap_h(103): 318,309,886,183,790,671,537,767,526, ( 108
    digits)
    745,028,724,068,919,291,480,912,897,
    495,334,688,117,793,595,268,474,246,
    168,599,718,714,334,570,984,041,935
    ------------------------------------------------------------------------------
    dap_k(103): 1, ( 109
    digits)
    000,000,000,000,000,000,000,000,000,
    000,000,000,000,000,000,000,000,000,
    000,000,000,000,000,000,000,000,000,
    000,000,000,000,000,000,000,000,000
    ------------------------------------------------------------------------------
    dividend(104): 16,968,701,043,152,952,636,467 ( 23
    digits)
    ------------------------------------------------------------------------------
    divisor(104): 15,819,637,988,916,675,843,163 ( 23
    digits)
    ------------------------------------------------------------------------------
    a(104): 1 ( 1
    digit )
    ------------------------------------------------------------------------------
    p(104): 29,746,719,516,721,832,313,949,402, ( 53
    digits)
    154,256,122,842,537,989,426,462,872
    ------------------------------------------------------------------------------
    q(104): 93,452,075,502,129,433,079,608,718, ( 53
    digits)
    805,973,266,139,117,968,309,671,261
    ------------------------------------------------------------------------------
    dap_h(104): 318,309,886,183,790,671,537,767,526, ( 108
    digits)
    745,028,724,068,919,291,480,912,897,
    495,334,688,117,793,595,268,474,246,
    168,599,718,714,334,570,984,041,802
    ------------------------------------------------------------------------------
    dap_k(104): 1, ( 109
    digits)
    000,000,000,000,000,000,000,000,000,
    000,000,000,000,000,000,000,000,000,
    000,000,000,000,000,000,000,000,000,
    000,000,000,000,000,000,000,000,000
    ------------------------------------------------------------------------------
    dividend(105): 15,819,637,988,916,675,843,163 ( 23
    digits)
    ------------------------------------------------------------------------------
    divisor(105): 1,149,063,054,236,276,793,304 ( 22
    digits)
    ------------------------------------------------------------------------------
    a(105): 13 ( 2
    digits)
    ------------------------------------------------------------------------------
    p(105): 412,308,791,148,657,435,430,497,399, ( 54
    digits)
    029,738,669,561,782,055,078,417,477
    ------------------------------------------------------------------------------
    q(105): 1, ( 55
    digits)
    295,306,269,283,110,567,140,456,863,
    215,662,145,871,184,342,734,796,918
    ------------------------------------------------------------------------------
    dap_h(105): 318,309,886,183,790,671,537,767,526, ( 108
    digits)
    745,028,724,068,919,291,480,912,897,
    495,334,688,117,793,595,268,474,246,
    168,599,718,714,334,570,984,041,810
    ------------------------------------------------------------------------------
    dap_k(105): 1, ( 109
    digits)
    000,000,000,000,000,000,000,000,000,
    000,000,000,000,000,000,000,000,000,
    000,000,000,000,000,000,000,000,000,
    000,000,000,000,000,000,000,000,000
    ------------------------------------------------------------------------------
    dividend(106): 1,149,063,054,236,276,793,304 ( 22
    digits)
    ------------------------------------------------------------------------------
    divisor(106): 881,818,283,845,077,530,211 ( 21
    digits)
    ------------------------------------------------------------------------------
    a(106): 1 ( 1
    digit )
    ------------------------------------------------------------------------------
    p(106): 442,055,510,665,379,267,744,446,801, ( 54
    digits)
    183,994,792,404,320,044,504,880,349
    ------------------------------------------------------------------------------
    q(106): 1, ( 55
    digits)
    388,758,344,785,240,000,220,065,582,
    021,635,412,010,302,311,044,468,179
    ------------------------------------------------------------------------------
    dap_h(106): 318,309,886,183,790,671,537,767,526, ( 108
    digits)
    745,028,724,068,919,291,480,912,897,
    495,334,688,117,793,595,268,474,246,
    168,599,718,714,334,570,984,041,810
    ------------------------------------------------------------------------------
    dap_k(106): 1, ( 109
    digits)
    000,000,000,000,000,000,000,000,000,
    000,000,000,000,000,000,000,000,000,
    000,000,000,000,000,000,000,000,000,
    000,000,000,000,000,000,000,000,000
    ------------------------------------------------------------------------------
    dividend(107): 881,818,283,845,077,530,211 ( 21
    digits)
    ------------------------------------------------------------------------------
    divisor(107): 267,244,770,391,199,263,093 ( 21
    digits)
    ------------------------------------------------------------------------------
    a(107): 3 ( 1
    digit )
    ------------------------------------------------------------------------------
    p(107): 1, ( 55
    digits)
    738,475,323,144,795,238,663,837,802,
    581,723,046,774,742,188,593,058,524
    ------------------------------------------------------------------------------
    q(107): 5, ( 55
    digits)
    461,581,303,638,830,567,800,653,609,
    280,568,381,902,091,275,868,201,455
    ------------------------------------------------------------------------------
    dap_h(107): 318,309,886,183,790,671,537,767,526, ( 108
    digits)
    745,028,724,068,919,291,480,912,897,
    495,334,688,117,793,595,268,474,246,
    168,599,718,714,334,570,984,041,810
    ------------------------------------------------------------------------------
    dap_k(107): 1, ( 109
    digits)
    000,000,000,000,000,000,000,000,000,
    000,000,000,000,000,000,000,000,000,
    000,000,000,000,000,000,000,000,000,
    000,000,000,000,000,000,000,000,000
    ------------------------------------------------------------------------------
    dividend(108): 267,244,770,391,199,263,093 ( 21
    digits)
    ------------------------------------------------------------------------------
    divisor(108): 80,083,972,671,479,740,932 ( 20
    digits)
    ------------------------------------------------------------------------------
    a(108): 3 ( 1
    digit )
    ------------------------------------------------------------------------------
    p(108): 5, ( 55
    digits)
    657,481,480,099,764,983,735,960,208,
    929,163,932,728,546,610,284,055,921
    ------------------------------------------------------------------------------
    q(108): 17, ( 56
    digits)
    773,502,255,701,731,703,622,026,409,
    863,340,557,716,576,138,649,072,544
    ------------------------------------------------------------------------------
    dap_h(108): 318,309,886,183,790,671,537,767,526, ( 108
    digits)
    745,028,724,068,919,291,480,912,897,
    495,334,688,117,793,595,268,474,246,
    168,599,718,714,334,570,984,041,810
    ------------------------------------------------------------------------------
    dap_k(108): 1, ( 109
    digits)
    000,000,000,000,000,000,000,000,000,
    000,000,000,000,000,000,000,000,000,
    000,000,000,000,000,000,000,000,000,
    000,000,000,000,000,000,000,000,000
    ------------------------------------------------------------------------------
    dividend(109): 80,083,972,671,479,740,932 ( 20
    digits)
    ------------------------------------------------------------------------------
    divisor(109): 26,992,852,376,760,040,297 ( 20
    digits)
    ------------------------------------------------------------------------------
    a(109): 2 ( 1
    digit )
    ------------------------------------------------------------------------------
    p(109): 13, ( 56
    digits)
    053,438,283,344,325,206,135,758,220,
    440,050,912,231,835,409,161,170,366
    ------------------------------------------------------------------------------
    q(109): 41, ( 56
    digits)
    008,585,815,042,293,975,044,706,429,
    007,249,497,335,243,553,166,346,543
    ------------------------------------------------------------------------------
    dap_h(109): 318,309,886,183,790,671,537,767,526, ( 108
    digits)
    745,028,724,068,919,291,480,912,897,
    495,334,688,117,793,595,268,474,246,
    168,599,718,714,334,570,984,041,810
    ------------------------------------------------------------------------------
    dap_k(109): 1, ( 109
    digits)
    000,000,000,000,000,000,000,000,000,
    000,000,000,000,000,000,000,000,000,
    000,000,000,000,000,000,000,000,000,
    000,000,000,000,000,000,000,000,000
    ------------------------------------------------------------------------------
    dividend(110): 26,992,852,376,760,040,297 ( 20
    digits)
    ------------------------------------------------------------------------------
    divisor(110): 26,098,267,917,959,660,338 ( 20
    digits)
    ------------------------------------------------------------------------------
    a(110): 1 ( 1
    digit )
    ------------------------------------------------------------------------------
    p(110): 18, ( 56
    digits)
    710,919,763,444,090,189,871,718,429,
    369,214,844,960,382,019,445,226,287
    ------------------------------------------------------------------------------
    q(110): 58, ( 56
    digits)
    782,088,070,744,025,678,666,732,838,
    870,590,055,051,819,691,815,419,087
    ------------------------------------------------------------------------------
    dap_h(110): 318,309,886,183,790,671,537,767,526, ( 108
    digits)
    745,028,724,068,919,291,480,912,897,
    495,334,688,117,793,595,268,474,246,
    168,599,718,714,334,570,984,041,810
    ------------------------------------------------------------------------------
    dap_k(110): 1, ( 109
    digits)
    000,000,000,000,000,000,000,000,000,
    000,000,000,000,000,000,000,000,000,
    000,000,000,000,000,000,000,000,000,
    000,000,000,000,000,000,000,000,000
    ------------------------------------------------------------------------------
    dividend(111): 26,098,267,917,959,660,338 ( 20
    digits)
    ------------------------------------------------------------------------------
    divisor(111): 894,584,458,800,379,959 ( 18
    digits)
    ------------------------------------------------------------------------------
    a(111): 29 ( 2
    digits)
    ------------------------------------------------------------------------------
    p(111): 555, ( 57
    digits)
    670,111,423,222,940,712,415,592,672,
    147,281,416,082,913,973,072,732,689
    ------------------------------------------------------------------------------
    q(111): 1,745, ( 58
    digits)
    689,139,866,619,038,656,379,958,756,
    254,361,093,838,014,615,813,500,066
    ------------------------------------------------------------------------------
    dap_h(111): 318,309,886,183,790,671,537,767,526, ( 108
    digits)
    745,028,724,068,919,291,480,912,897,
    495,334,688,117,793,595,268,474,246,
    168,599,718,714,334,570,984,041,810
    ------------------------------------------------------------------------------
    dap_k(111): 1, ( 109
    digits)
    000,000,000,000,000,000,000,000,000,
    000,000,000,000,000,000,000,000,000,
    000,000,000,000,000,000,000,000,000,
    000,000,000,000,000,000,000,000,000
    ------------------------------------------------------------------------------
    dividend(112): 894,584,458,800,379,959 ( 18
    digits)
    ------------------------------------------------------------------------------
    divisor(112): 155,318,612,748,641,527 ( 18
    digits)
    ------------------------------------------------------------------------------
    a(112): 5 ( 1
    digit )
    ------------------------------------------------------------------------------
    p(112): 2,797, ( 58
    digits)
    061,476,879,558,793,751,949,681,790,
    105,621,925,374,951,884,808,889,732
    ------------------------------------------------------------------------------
    q(112): 8,787, ( 58
    digits)
    227,787,403,839,218,960,566,526,620,
    142,395,524,241,892,770,882,919,417
    ------------------------------------------------------------------------------
    dap_h(112): 318,309,886,183,790,671,537,767,526, ( 108
    digits)
    745,028,724,068,919,291,480,912,897,
    495,334,688,117,793,595,268,474,246,
    168,599,718,714,334,570,984,041,810
    ------------------------------------------------------------------------------
    dap_k(112): 1, ( 109
    digits)
    000,000,000,000,000,000,000,000,000,
    000,000,000,000,000,000,000,000,000,
    000,000,000,000,000,000,000,000,000,
    000,000,000,000,000,000,000,000,000
    ------------------------------------------------------------------------------
    dividend(113): 155,318,612,748,641,527 ( 18
    digits)
    ------------------------------------------------------------------------------
    divisor(113): 117,991,395,057,172,324 ( 18
    digits)
    ------------------------------------------------------------------------------
    a(113): 1 ( 1
    digit )
    ------------------------------------------------------------------------------
    p(113): 3,352, ( 58
    digits)
    731,588,302,781,734,464,365,274,462,
    252,903,341,457,865,857,881,622,421
    ------------------------------------------------------------------------------
    q(113): 10,532, ( 59
    digits)
    916,927,270,458,257,616,946,485,376,
    396,756,618,079,907,386,696,419,483
    ------------------------------------------------------------------------------
    dap_h(113): 318,309,886,183,790,671,537,767,526, ( 108
    digits)
    745,028,724,068,919,291,480,912,897,
    495,334,688,117,793,595,268,474,246,
    168,599,718,714,334,570,984,041,810
    ------------------------------------------------------------------------------
    dap_k(113): 1, ( 109
    digits)
    000,000,000,000,000,000,000,000,000,
    000,000,000,000,000,000,000,000,000,
    000,000,000,000,000,000,000,000,000,
    000,000,000,000,000,000,000,000,000
    ------------------------------------------------------------------------------
    dividend(114): 117,991,395,057,172,324 ( 18
    digits)
    ------------------------------------------------------------------------------
    divisor(114): 37,327,217,691,469,203 ( 17
    digits)
    ------------------------------------------------------------------------------
    a(114): 3 ( 1
    digit )
    ------------------------------------------------------------------------------
    p(114): 12,855, ( 59
    digits)
    256,241,787,903,997,145,045,505,176,
    864,331,949,748,549,458,453,756,995
    ------------------------------------------------------------------------------
    q(114): 40,385, ( 59
    digits)
    978,569,215,213,991,811,405,982,749,
    332,665,378,481,614,930,972,177,866
    ------------------------------------------------------------------------------
    dap_h(114): 318,309,886,183,790,671,537,767,526, ( 108
    digits)
    745,028,724,068,919,291,480,912,897,
    495,334,688,117,793,595,268,474,246,
    168,599,718,714,334,570,984,041,810
    ------------------------------------------------------------------------------
    dap_k(114): 1, ( 109
    digits)
    000,000,000,000,000,000,000,000,000,
    000,000,000,000,000,000,000,000,000,
    000,000,000,000,000,000,000,000,000,
    000,000,000,000,000,000,000,000,000
    ------------------------------------------------------------------------------
    dividend(115): 37,327,217,691,469,203 ( 17
    digits)
    ------------------------------------------------------------------------------
    divisor(115): 6,009,741,982,764,715 ( 16
    digits)
    ------------------------------------------------------------------------------
    a(115): 6 ( 1
    digit )
    ------------------------------------------------------------------------------
    p(115): 80,484, ( 59
    digits)
    269,039,030,205,717,334,638,305,523,
    438,895,039,949,162,608,604,164,391
    ------------------------------------------------------------------------------
    q(115): 252,848, ( 60
    digits)
    788,342,561,742,208,485,382,381,872,
    392,748,888,969,596,972,529,486,679
    ------------------------------------------------------------------------------
    dap_h(115): 318,309,886,183,790,671,537,767,526, ( 108
    digits)
    745,028,724,068,919,291,480,912,897,
    495,334,688,117,793,595,268,474,246,
    168,599,718,714,334,570,984,041,810
    ------------------------------------------------------------------------------
    dap_k(115): 1, ( 109
    digits)
    000,000,000,000,000,000,000,000,000,
    000,000,000,000,000,000,000,000,000,
    000,000,000,000,000,000,000,000,000,
    000,000,000,000,000,000,000,000,000
    ------------------------------------------------------------------------------
    dividend(116): 6,009,741,982,764,715 ( 16
    digits)
    ------------------------------------------------------------------------------
    divisor(116): 1,268,765,794,880,913 ( 16
    digits)
    ------------------------------------------------------------------------------
    a(116): 4 ( 1
    digit )
    ------------------------------------------------------------------------------
    p(116): 334,792, ( 60
    digits)
    332,397,908,726,866,483,598,727,270,
    619,912,109,545,199,892,870,414,559
    ------------------------------------------------------------------------------
    q(116): 1,051,781, ( 61
    digits)
    131,939,462,182,825,752,935,510,238,
    903,660,934,360,002,821,090,124,582
    ------------------------------------------------------------------------------
    dap_h(116): 318,309,886,183,790,671,537,767,526, ( 108
    digits)
    745,028,724,068,919,291,480,912,897,
    495,334,688,117,793,595,268,474,246,
    168,599,718,714,334,570,984,041,810
    ------------------------------------------------------------------------------
    dap_k(116): 1, ( 109
    digits)
    000,000,000,000,000,000,000,000,000,
    000,000,000,000,000,000,000,000,000,
    000,000,000,000,000,000,000,000,000,
    000,000,000,000,000,000,000,000,000
    ------------------------------------------------------------------------------
    dividend(117): 1,268,765,794,880,913 ( 16
    digits)
    ------------------------------------------------------------------------------
    divisor(117): 934,678,803,241,063 ( 15
    digits)
    ------------------------------------------------------------------------------
    a(117): 1 ( 1
    digit )
    ------------------------------------------------------------------------------
    p(117): 415,276, ( 60
    digits)
    601,436,938,932,583,818,237,032,794,
    058,807,149,494,362,501,474,578,950
    ------------------------------------------------------------------------------
    q(117): 1,304,629, ( 61
    digits)
    920,282,023,925,034,238,317,892,111,
    296,409,823,329,599,793,619,611,261
    ------------------------------------------------------------------------------
    dap_h(117): 318,309,886,183,790,671,537,767,526, ( 108
    digits)
    745,028,724,068,919,291,480,912,897,
    495,334,688,117,793,595,268,474,246,
    168,599,718,714,334,570,984,041,810
    ------------------------------------------------------------------------------
    dap_k(117): 1, ( 109
    digits)
    000,000,000,000,000,000,000,000,000,
    000,000,000,000,000,000,000,000,000,
    000,000,000,000,000,000,000,000,000,
    000,000,000,000,000,000,000,000,000
    ------------------------------------------------------------------------------
    dividend(118): 934,678,803,241,063 ( 15
    digits)
    ------------------------------------------------------------------------------
    divisor(118): 334,086,991,639,850 ( 15
    digits)
    ------------------------------------------------------------------------------
    a(118): 2 ( 1
    digit )
    ------------------------------------------------------------------------------
    p(118): 1,165,345, ( 61
    digits)
    535,271,786,592,034,120,072,792,858,
    737,526,408,533,924,895,819,572,459
    ------------------------------------------------------------------------------
    q(118): 3,661,040, ( 61
    digits)
    972,503,510,032,894,229,571,294,461,
    496,480,581,019,202,408,329,347,104
    ------------------------------------------------------------------------------
    dap_h(118): 318,309,886,183,790,671,537,767,526, ( 108
    digits)
    745,028,724,068,919,291,480,912,897,
    495,334,688,117,793,595,268,474,246,
    168,599,718,714,334,570,984,041,810
    ------------------------------------------------------------------------------
    dap_k(118): 1, ( 109
    digits)
    000,000,000,000,000,000,000,000,000,
    000,000,000,000,000,000,000,000,000,
    000,000,000,000,000,000,000,000,000,
    000,000,000,000,000,000,000,000,000
    ------------------------------------------------------------------------------
    dividend(119): 334,086,991,639,850 ( 15
    digits)
    ------------------------------------------------------------------------------
    divisor(119): 266,504,819,961,363 ( 15
    digits)
    ------------------------------------------------------------------------------
    a(119): 1 ( 1
    digit )
    ------------------------------------------------------------------------------
    p(119): 1,580,622, ( 61
    digits)
    136,708,725,524,617,938,309,825,652,
    796,333,558,028,287,397,294,151,409
    ------------------------------------------------------------------------------
    q(119): 4,965,670, ( 61
    digits)
    892,785,533,957,928,467,889,186,572,
    792,890,404,348,802,201,948,958,365
    ------------------------------------------------------------------------------
    dap_h(119): 318,309,886,183,790,671,537,767,526, ( 108
    digits)
    745,028,724,068,919,291,480,912,897,
    495,334,688,117,793,595,268,474,246,
    168,599,718,714,334,570,984,041,810
    ------------------------------------------------------------------------------
    dap_k(119): 1, ( 109
    digits)
    000,000,000,000,000,000,000,000,000,
    000,000,000,000,000,000,000,000,000,
    000,000,000,000,000,000,000,000,000,
    000,000,000,000,000,000,000,000,000
    ------------------------------------------------------------------------------
    dividend(120): 266,504,819,961,363 ( 15
    digits)
    ------------------------------------------------------------------------------
    divisor(120): 67,582,171,678,487 ( 14
    digits)
    ------------------------------------------------------------------------------
    a(120): 3 ( 1
    digit )
    ------------------------------------------------------------------------------
    p(120): 5,907,211, ( 61
    digits)
    945,397,963,165,887,935,002,269,817,
    126,527,082,618,787,087,702,026,686
    ------------------------------------------------------------------------------
    q(120): 18,558,053, ( 62
    digits)
    650,860,111,906,679,633,238,854,179,
    875,151,794,065,609,014,176,222,199
    ------------------------------------------------------------------------------
    dap_h(120): 318,309,886,183,790,671,537,767,526, ( 108
    digits)
    745,028,724,068,919,291,480,912,897,
    495,334,688,117,793,595,268,474,246,
    168,599,718,714,334,570,984,041,810
    ------------------------------------------------------------------------------
    dap_k(120): 1, ( 109
    digits)
    000,000,000,000,000,000,000,000,000,
    000,000,000,000,000,000,000,000,000,
    000,000,000,000,000,000,000,000,000,
    000,000,000,000,000,000,000,000,000
    ------------------------------------------------------------------------------
    dividend(121): 67,582,171,678,487 ( 14
    digits)
    ------------------------------------------------------------------------------
    divisor(121): 63,758,304,925,902 ( 14
    digits)
    ------------------------------------------------------------------------------
    a(121): 1 ( 1
    digit )
    ------------------------------------------------------------------------------
    p(121): 7,487,834, ( 61
    digits)
    082,106,688,690,505,873,312,095,469,
    922,860,640,647,074,484,996,178,095
    ------------------------------------------------------------------------------
    q(121): 23,523,724, ( 62
    digits)
    543,645,645,864,608,101,128,040,752,
    668,042,198,414,411,216,125,180,564
    ------------------------------------------------------------------------------
    dap_h(121): 318,309,886,183,790,671,537,767,526, ( 108
    digits)
    745,028,724,068,919,291,480,912,897,
    495,334,688,117,793,595,268,474,246,
    168,599,718,714,334,570,984,041,810
    ------------------------------------------------------------------------------
    dap_k(121): 1, ( 109
    digits)
    000,000,000,000,000,000,000,000,000,
    000,000,000,000,000,000,000,000,000,
    000,000,000,000,000,000,000,000,000,
    000,000,000,000,000,000,000,000,000
    ------------------------------------------------------------------------------
    dividend(122): 63,758,304,925,902 ( 14
    digits)
    ------------------------------------------------------------------------------
    divisor(122): 3,823,866,752,585 ( 13
    digits)
    ------------------------------------------------------------------------------
    a(122): 16 ( 2
    digits)
    ------------------------------------------------------------------------------
    p(122): 125,712,557, ( 63
    digits)
    259,104,982,213,981,907,995,797,335,
    892,297,332,971,978,847,640,876,206
    ------------------------------------------------------------------------------
    q(122): 394,937,646, ( 63
    digits)
    349,190,445,740,409,251,287,506,222,
    563,826,968,696,188,472,179,111,223
    ------------------------------------------------------------------------------
    dap_h(122): 318,309,886,183,790,671,537,767,526, ( 108
    digits)
    745,028,724,068,919,291,480,912,897,
    495,334,688,117,793,595,268,474,246,
    168,599,718,714,334,570,984,041,810
    ------------------------------------------------------------------------------
    dap_k(122): 1, ( 109
    digits)
    000,000,000,000,000,000,000,000,000,
    000,000,000,000,000,000,000,000,000,
    000,000,000,000,000,000,000,000,000,
    000,000,000,000,000,000,000,000,000
    ------------------------------------------------------------------------------
    dividend(123): 3,823,866,752,585 ( 13
    digits)
    ------------------------------------------------------------------------------
    divisor(123): 2,576,436,884,542 ( 13
    digits)
    ------------------------------------------------------------------------------
    a(123): 1 ( 1
    digit )
    ------------------------------------------------------------------------------
    p(123): 133,200,391, ( 63
    digits)
    341,211,670,904,487,781,307,892,805,
    815,157,973,619,053,332,637,054,301
    ------------------------------------------------------------------------------
    q(123): 418,461,370, ( 63
    digits)
    892,836,091,605,017,352,415,546,975,
    231,869,167,110,599,688,304,291,787
    ------------------------------------------------------------------------------
    dap_h(123): 318,309,886,183,790,671,537,767,526, ( 108
    digits)
    745,028,724,068,919,291,480,912,897,
    495,334,688,117,793,595,268,474,246,
    168,599,718,714,334,570,984,041,810
    ------------------------------------------------------------------------------
    dap_k(123): 1, ( 109
    digits)
    000,000,000,000,000,000,000,000,000,
    000,000,000,000,000,000,000,000,000,
    000,000,000,000,000,000,000,000,000,
    000,000,000,000,000,000,000,000,000
    ------------------------------------------------------------------------------
    dividend(124): 2,576,436,884,542 ( 13
    digits)
    ------------------------------------------------------------------------------
    divisor(124): 1,247,429,868,043 ( 13
    digits)
    ------------------------------------------------------------------------------
    a(124): 2 ( 1
    digit )
    ------------------------------------------------------------------------------
    p(124): 392,113,339, ( 63
    digits)
    941,528,324,022,957,470,611,582,947,
    522,613,280,210,085,512,914,984,808
    ------------------------------------------------------------------------------
    q(124): 1,231,860,388, ( 64
    digits)
    134,862,628,950,443,956,118,600,173,
    027,565,302,917,387,848,787,694,797
    ------------------------------------------------------------------------------
    dap_h(124): 318,309,886,183,790,671,537,767,526, ( 108
    digits)
    745,028,724,068,919,291,480,912,897,
    495,334,688,117,793,595,268,474,246,
    168,599,718,714,334,570,984,041,810
    ------------------------------------------------------------------------------
    dap_k(124): 1, ( 109
    digits)
    000,000,000,000,000,000,000,000,000,
    000,000,000,000,000,000,000,000,000,
    000,000,000,000,000,000,000,000,000,
    000,000,000,000,000,000,000,000,000
    ------------------------------------------------------------------------------
    dividend(125): 1,247,429,868,043 ( 13
    digits)
    ------------------------------------------------------------------------------
    divisor(125): 81,577,148,456 ( 11
    digits)
    ------------------------------------------------------------------------------
    a(125): 15 ( 2
    digits)
    ------------------------------------------------------------------------------
    p(125): 6,014,900,490, ( 64
    digits)
    464,136,531,248,849,840,481,637,018,
    654,357,176,770,336,026,361,826,421
    ------------------------------------------------------------------------------
    q(125): 18,896,367,192, ( 65
    digits)
    915,775,525,861,676,694,194,549,570,
    645,348,710,871,417,420,119,713,742
    ------------------------------------------------------------------------------
    dap_h(125): 318,309,886,183,790,671,537,767,526, ( 108
    digits)
    745,028,724,068,919,291,480,912,897,
    495,334,688,117,793,595,268,474,246,
    168,599,718,714,334,570,984,041,810
    ------------------------------------------------------------------------------
    dap_k(125): 1, ( 109
    digits)
    000,000,000,000,000,000,000,000,000,
    000,000,000,000,000,000,000,000,000,
    000,000,000,000,000,000,000,000,000,
    000,000,000,000,000,000,000,000,000
    ------------------------------------------------------------------------------
    dividend(126): 81,577,148,456 ( 11
    digits)
    ------------------------------------------------------------------------------
    divisor(126): 23,772,641,203 ( 11
    digits)
    ------------------------------------------------------------------------------
    a(126): 3 ( 1
    digit )
    ------------------------------------------------------------------------------
    p(126): 18,436,814,811, ( 65
    digits)
    333,937,917,769,506,992,056,494,003,
    485,684,810,521,093,592,000,464,071
    ------------------------------------------------------------------------------
    q(126): 57,920,961,966, ( 65
    digits)
    882,189,206,535,474,038,702,248,884,
    963,611,435,531,640,109,146,836,023
    ------------------------------------------------------------------------------
    dap_h(126): 318,309,886,183,790,671,537,767,526, ( 108
    digits)
    745,028,724,068,919,291,480,912,897,
    495,334,688,117,793,595,268,474,246,
    168,599,718,714,334,570,984,041,810
    ------------------------------------------------------------------------------
    dap_k(126): 1, ( 109
    digits)
    000,000,000,000,000,000,000,000,000,
    000,000,000,000,000,000,000,000,000,
    000,000,000,000,000,000,000,000,000,
    000,000,000,000,000,000,000,000,000
    ------------------------------------------------------------------------------
    dividend(127): 23,772,641,203 ( 11
    digits)
    ------------------------------------------------------------------------------
    divisor(127): 10,259,224,847 ( 11
    digits)
    ------------------------------------------------------------------------------
    a(127): 2 ( 1
    digit )
    ------------------------------------------------------------------------------
    p(127): 42,888,530,113, ( 65
    digits)
    132,012,366,787,863,824,594,625,025,
    625,726,797,812,523,210,362,754,563
    ------------------------------------------------------------------------------
    q(127): 134,738,291,126, ( 66
    digits)
    680,153,938,932,624,771,599,047,340,
    572,571,581,934,697,638,413,385,788
    ------------------------------------------------------------------------------
    dap_h(127): 318,309,886,183,790,671,537,767,526, ( 108
    digits)
    745,028,724,068,919,291,480,912,897,
    495,334,688,117,793,595,268,474,246,
    168,599,718,714,334,570,984,041,810
    ------------------------------------------------------------------------------
    dap_k(127): 1, ( 109
    digits)
    000,000,000,000,000,000,000,000,000,
    000,000,000,000,000,000,000,000,000,
    000,000,000,000,000,000,000,000,000,
    000,000,000,000,000,000,000,000,000
    ------------------------------------------------------------------------------
    dividend(128): 10,259,224,847 ( 11
    digits)
    ------------------------------------------------------------------------------
    divisor(128): 3,254,191,509 ( 10
    digits)
    ------------------------------------------------------------------------------
    a(128): 3 ( 1
    digit )
    ------------------------------------------------------------------------------
    p(128): 147,102,405,150, ( 66
    digits)
    729,975,018,133,098,465,840,369,080,
    362,865,203,958,663,223,088,727,760
    ------------------------------------------------------------------------------
    q(128): 462,135,835,346, ( 66
    digits)
    922,651,023,333,348,353,499,390,906,
    681,326,181,335,733,024,386,993,387
    ------------------------------------------------------------------------------
    dap_h(128): 318,309,886,183,790,671,537,767,526, ( 108
    digits)
    745,028,724,068,919,291,480,912,897,
    495,334,688,117,793,595,268,474,246,
    168,599,718,714,334,570,984,041,810
    ------------------------------------------------------------------------------
    dap_k(128): 1, ( 109
    digits)
    000,000,000,000,000,000,000,000,000,
    000,000,000,000,000,000,000,000,000,
    000,000,000,000,000,000,000,000,000,
    000,000,000,000,000,000,000,000,000
    ------------------------------------------------------------------------------
    dividend(129): 3,254,191,509 ( 10
    digits)
    ------------------------------------------------------------------------------
    divisor(129): 496,650,320 ( 9
    digits)
    ------------------------------------------------------------------------------
    a(129): 6 ( 1
    digit )
    ------------------------------------------------------------------------------
    p(129): 925,502,961,017, ( 66
    digits)
    511,862,475,586,454,619,636,839,507,
    802,918,021,564,502,548,895,121,123
    ------------------------------------------------------------------------------
    q(129): 2,907,553,303,208, ( 67
    digits)
    216,060,078,932,714,892,595,392,780,
    660,528,669,949,095,784,735,346,110
    ------------------------------------------------------------------------------
    dap_h(129): 318,309,886,183,790,671,537,767,526, ( 108
    digits)
    745,028,724,068,919,291,480,912,897,
    495,334,688,117,793,595,268,474,246,
    168,599,718,714,334,570,984,041,810
    ------------------------------------------------------------------------------
    dap_k(129): 1, ( 109
    digits)
    000,000,000,000,000,000,000,000,000,
    000,000,000,000,000,000,000,000,000,
    000,000,000,000,000,000,000,000,000,
    000,000,000,000,000,000,000,000,000
    ------------------------------------------------------------------------------
    dividend(130): 496,650,320 ( 9
    digits)
    ------------------------------------------------------------------------------
    divisor(130): 274,289,589 ( 9
    digits)
    ------------------------------------------------------------------------------
    a(130): 1 ( 1
    digit )
    ------------------------------------------------------------------------------
    p(130): 1,072,605,366,168, ( 67
    digits)
    241,837,493,719,553,085,477,208,588,
    165,783,225,523,165,771,983,848,883
    ------------------------------------------------------------------------------
    q(130): 3,369,689,138,555, ( 67
    digits)
    138,711,102,266,063,246,094,783,687,
    341,854,851,284,828,809,122,339,497
    ------------------------------------------------------------------------------
    dap_h(130): 318,309,886,183,790,671,537,767,526, ( 108
    digits)
    745,028,724,068,919,291,480,912,897,
    495,334,688,117,793,595,268,474,246,
    168,599,718,714,334,570,984,041,810
    ------------------------------------------------------------------------------
    dap_k(130): 1, ( 109
    digits)
    000,000,000,000,000,000,000,000,000,
    000,000,000,000,000,000,000,000,000,
    000,000,000,000,000,000,000,000,000,
    000,000,000,000,000,000,000,000,000
    ------------------------------------------------------------------------------
    dividend(131): 274,289,589 ( 9
    digits)
    ------------------------------------------------------------------------------
    divisor(131): 222,360,731 ( 9
    digits)
    ------------------------------------------------------------------------------
    a(131): 1 ( 1
    digit )
    ------------------------------------------------------------------------------
    p(131): 1,998,108,327,185, ( 67
    digits)
    753,699,969,306,007,705,114,048,095,
    968,701,247,087,668,320,878,970,006
    ------------------------------------------------------------------------------
    q(131): 6,277,242,441,763, ( 67
    digits)
    354,771,181,198,778,138,690,176,468,
    002,383,521,233,924,593,857,685,607
    ------------------------------------------------------------------------------
    dap_h(131): 318,309,886,183,790,671,537,767,526, ( 108
    digits)
    745,028,724,068,919,291,480,912,897,
    495,334,688,117,793,595,268,474,246,
    168,599,718,714,334,570,984,041,810
    ------------------------------------------------------------------------------
    dap_k(131): 1, ( 109
    digits)
    000,000,000,000,000,000,000,000,000,
    000,000,000,000,000,000,000,000,000,
    000,000,000,000,000,000,000,000,000,
    000,000,000,000,000,000,000,000,000
    ------------------------------------------------------------------------------
    dividend(132): 222,360,731 ( 9
    digits)
    ------------------------------------------------------------------------------
    divisor(132): 51,928,858 ( 8
    digits)
    ------------------------------------------------------------------------------
    a(132): 4 ( 1
    digit )
    ------------------------------------------------------------------------------
    p(132): 9,065,038,674,911, ( 67
    digits)
    256,637,370,943,583,905,933,400,972,
    040,588,213,873,839,055,499,728,907
    ------------------------------------------------------------------------------
    q(132): 28,478,658,905,608, ( 68
    digits)
    557,795,827,061,175,800,855,489,559,
    351,388,936,220,527,184,553,081,925
    ------------------------------------------------------------------------------
    dap_h(132): 318,309,886,183,790,671,537,767,526, ( 108
    digits)
    745,028,724,068,919,291,480,912,897,
    495,334,688,117,793,595,268,474,246,
    168,599,718,714,334,570,984,041,810
    ------------------------------------------------------------------------------
    dap_k(132): 1, ( 109
    digits)
    000,000,000,000,000,000,000,000,000,
    000,000,000,000,000,000,000,000,000,
    000,000,000,000,000,000,000,000,000,
    000,000,000,000,000,000,000,000,000
    ------------------------------------------------------------------------------
    dividend(133): 51,928,858 ( 8
    digits)
    ------------------------------------------------------------------------------
    divisor(133): 14,645,299 ( 8
    digits)
    ------------------------------------------------------------------------------
    a(133): 3 ( 1
    digit )
    ------------------------------------------------------------------------------
    p(133): 29,193,224,351,919, ( 68
    digits)
    523,612,082,136,759,422,914,251,012,
    090,465,888,709,185,487,378,156,727
    ------------------------------------------------------------------------------
    q(133): 91,713,219,158,589, ( 68
    digits)
    028,158,662,382,305,541,256,645,146,
    056,550,329,895,506,147,516,931,382
    ------------------------------------------------------------------------------
    dap_h(133): 318,309,886,183,790,671,537,767,526, ( 108
    digits)
    745,028,724,068,919,291,480,912,897,
    495,334,688,117,793,595,268,474,246,
    168,599,718,714,334,570,984,041,810
    ------------------------------------------------------------------------------
    dap_k(133): 1, ( 109
    digits)
    000,000,000,000,000,000,000,000,000,
    000,000,000,000,000,000,000,000,000,
    000,000,000,000,000,000,000,000,000,
    000,000,000,000,000,000,000,000,000
    ------------------------------------------------------------------------------
    dividend(134): 14,645,299 ( 8
    digits)
    ------------------------------------------------------------------------------
    divisor(134): 7,992,961 ( 7
    digits)
    ------------------------------------------------------------------------------
    a(134): 1 ( 1
    digit )
    ------------------------------------------------------------------------------
    p(134): 38,258,263,026,830, ( 68
    digits)
    780,249,453,080,343,328,847,651,984,
    131,054,102,583,024,542,877,885,634
    ------------------------------------------------------------------------------
    q(134): 120,191,878,064,197, ( 69
    digits)
    585,954,489,443,481,342,112,134,705,
    407,939,266,116,033,332,070,013,307
    ------------------------------------------------------------------------------
    dap_h(134): 318,309,886,183,790,671,537,767,526, ( 108
    digits)
    745,028,724,068,919,291,480,912,897,
    495,334,688,117,793,595,268,474,246,
    168,599,718,714,334,570,984,041,810
    ------------------------------------------------------------------------------
    dap_k(134): 1, ( 109
    digits)
    000,000,000,000,000,000,000,000,000,
    000,000,000,000,000,000,000,000,000,
    000,000,000,000,000,000,000,000,000,
    000,000,000,000,000,000,000,000,000
    ------------------------------------------------------------------------------
    dividend(135): 7,992,961 ( 7
    digits)
    ------------------------------------------------------------------------------
    divisor(135): 6,652,338 ( 7
    digits)
    ------------------------------------------------------------------------------
    a(135): 1 ( 1
    digit )
    ------------------------------------------------------------------------------
    p(135): 67,451,487,378,750, ( 68
    digits)
    303,861,535,217,102,751,761,902,996,
    221,519,991,292,210,030,256,042,361
    ------------------------------------------------------------------------------
    q(135): 211,905,097,222,786, ( 69
    digits)
    614,113,151,825,786,883,368,779,851,
    464,489,596,011,539,479,586,944,689
    ------------------------------------------------------------------------------
    dap_h(135): 318,309,886,183,790,671,537,767,526, ( 108
    digits)
    745,028,724,068,919,291,480,912,897,
    495,334,688,117,793,595,268,474,246,
    168,599,718,714,334,570,984,041,810
    ------------------------------------------------------------------------------
    dap_k(135): 1, ( 109
    digits)
    000,000,000,000,000,000,000,000,000,
    000,000,000,000,000,000,000,000,000,
    000,000,000,000,000,000,000,000,000,
    000,000,000,000,000,000,000,000,000
    ------------------------------------------------------------------------------
    dividend(136): 6,652,338 ( 7
    digits)
    ------------------------------------------------------------------------------
    divisor(136): 1,340,623 ( 7
    digits)
    ------------------------------------------------------------------------------
    a(136): 4 ( 1
    digit )
    ------------------------------------------------------------------------------
    p(136): 308,064,212,541,831, ( 69
    digits)
    995,695,593,948,754,335,895,263,969,
    017,134,067,751,864,663,902,055,078
    ------------------------------------------------------------------------------
    q(136): 967,812,266,955,344, ( 69
    digits)
    042,407,096,746,628,875,587,254,111,
    265,897,650,162,191,250,417,792,063
    ------------------------------------------------------------------------------
    dap_h(136): 318,309,886,183,790,671,537,767,526, ( 108
    digits)
    745,028,724,068,919,291,480,912,897,
    495,334,688,117,793,595,268,474,246,
    168,599,718,714,334,570,984,041,810
    ------------------------------------------------------------------------------
    dap_k(136): 1, ( 109
    digits)
    000,000,000,000,000,000,000,000,000,
    000,000,000,000,000,000,000,000,000,
    000,000,000,000,000,000,000,000,000,
    000,000,000,000,000,000,000,000,000
    ------------------------------------------------------------------------------
    dividend(137): 1,340,623 ( 7
    digits)
    ------------------------------------------------------------------------------
    divisor(137): 1,289,846 ( 7
    digits)
    ------------------------------------------------------------------------------
    a(137): 1 ( 1
    digit )
    ------------------------------------------------------------------------------
    p(137): 375,515,699,920,582, ( 69
    digits)
    299,557,129,165,857,087,657,166,965,
    238,654,059,044,074,694,158,097,439
    ------------------------------------------------------------------------------
    q(137): 1,179,717,364,178,130, ( 70
    digits)
    656,520,248,572,415,758,956,033,962,
    730,387,246,173,730,730,004,736,752
    ------------------------------------------------------------------------------
    dap_h(137): 318,309,886,183,790,671,537,767,526, ( 108
    digits)
    745,028,724,068,919,291,480,912,897,
    495,334,688,117,793,595,268,474,246,
    168,599,718,714,334,570,984,041,810
    ------------------------------------------------------------------------------
    dap_k(137): 1, ( 109
    digits)
    000,000,000,000,000,000,000,000,000,
    000,000,000,000,000,000,000,000,000,
    000,000,000,000,000,000,000,000,000,
    000,000,000,000,000,000,000,000,000
    ------------------------------------------------------------------------------
    dividend(138): 1,289,846 ( 7
    digits)
    ------------------------------------------------------------------------------
    divisor(138): 50,777 ( 5
    digits)
    ------------------------------------------------------------------------------
    a(138): 25 ( 2
    digits)
    ------------------------------------------------------------------------------
    p(138): 9,695,956,710,556,389, ( 70
    digits)
    484,623,823,095,181,527,324,438,099,
    983,485,543,853,732,017,854,491,053
    ------------------------------------------------------------------------------
    q(138): 30,460,746,371,408,610, ( 71
    digits)
    455,413,311,057,022,849,488,103,179,
    525,578,804,505,459,500,536,210,863
    ------------------------------------------------------------------------------
    dap_h(138): 318,309,886,183,790,671,537,767,526, ( 108
    digits)
    745,028,724,068,919,291,480,912,897,
    495,334,688,117,793,595,268,474,246,
    168,599,718,714,334,570,984,041,810
    ------------------------------------------------------------------------------
    dap_k(138): 1, ( 109
    digits)
    000,000,000,000,000,000,000,000,000,
    000,000,000,000,000,000,000,000,000,
    000,000,000,000,000,000,000,000,000,
    000,000,000,000,000,000,000,000,000
    ------------------------------------------------------------------------------
    dividend(139): 50,777 ( 5
    digits)
    ------------------------------------------------------------------------------
    divisor(139): 20,421 ( 5
    digits)
    ------------------------------------------------------------------------------
    a(139): 2 ( 1
    digit )
    ------------------------------------------------------------------------------
    p(139): 19,767,429,121,033,361, ( 71
    digits)
    268,804,775,356,220,142,306,043,165,
    205,625,146,751,538,729,867,079,545
    ------------------------------------------------------------------------------
    q(139): 62,101,210,106,995,351, ( 71
    digits)
    567,346,870,686,461,457,932,240,321,
    781,544,855,184,649,731,077,158,478
    ------------------------------------------------------------------------------
    dap_h(139): 318,309,886,183,790,671,537,767,526, ( 108
    digits)
    745,028,724,068,919,291,480,912,897,
    495,334,688,117,793,595,268,474,246,
    168,599,718,714,334,570,984,041,810
    ------------------------------------------------------------------------------
    dap_k(139): 1, ( 109
    digits)
    000,000,000,000,000,000,000,000,000,
    000,000,000,000,000,000,000,000,000,
    000,000,000,000,000,000,000,000,000,
    000,000,000,000,000,000,000,000,000
    ------------------------------------------------------------------------------
    dividend(140): 20,421 ( 5
    digits)
    ------------------------------------------------------------------------------
    divisor(140): 9,935 ( 4
    digits)
    ------------------------------------------------------------------------------
    a(140): 2 ( 1
    digit )
    ------------------------------------------------------------------------------
    p(140): 49,230,814,952,623,112, ( 71
    digits)
    022,233,373,807,621,811,936,524,430,
    394,735,837,356,809,477,588,650,143
    ------------------------------------------------------------------------------
    q(140): 154,663,166,585,399,313, ( 72
    digits)
    590,107,052,429,945,765,352,583,823,
    088,668,514,874,758,962,690,527,819
    ------------------------------------------------------------------------------
    dap_h(140): 318,309,886,183,790,671,537,767,526, ( 108
    digits)
    745,028,724,068,919,291,480,912,897,
    495,334,688,117,793,595,268,474,246,
    168,599,718,714,334,570,984,041,810
    ------------------------------------------------------------------------------
    dap_k(140): 1, ( 109
    digits)
    000,000,000,000,000,000,000,000,000,
    000,000,000,000,000,000,000,000,000,
    000,000,000,000,000,000,000,000,000,
    000,000,000,000,000,000,000,000,000
    ------------------------------------------------------------------------------
    dividend(141): 9,935 ( 4
    digits)
    ------------------------------------------------------------------------------
    divisor(141): 551 ( 3
    digits)
    ------------------------------------------------------------------------------
    a(141): 18 ( 2
    digits)
    ------------------------------------------------------------------------------
    p(141): 905,922,098,268,249,377, ( 72
    digits)
    669,005,503,893,412,757,163,482,912,
    310,870,219,174,109,326,462,782,119
    ------------------------------------------------------------------------------
    q(141): 2,846,038,208,644,182,996, ( 73
    digits)
    189,273,814,425,485,234,278,749,137,
    377,578,122,930,311,059,506,659,220
    ------------------------------------------------------------------------------
    dap_h(141): 318,309,886,183,790,671,537,767,526, ( 108
    digits)
    745,028,724,068,919,291,480,912,897,
    495,334,688,117,793,595,268,474,246,
    168,599,718,714,334,570,984,041,810
    ------------------------------------------------------------------------------
    dap_k(141): 1, ( 109
    digits)
    000,000,000,000,000,000,000,000,000,
    000,000,000,000,000,000,000,000,000,
    000,000,000,000,000,000,000,000,000,
    000,000,000,000,000,000,000,000,000
    ------------------------------------------------------------------------------
    dividend(142): 551 ( 3
    digits)
    ------------------------------------------------------------------------------
    divisor(142): 17 ( 2
    digits)
    ------------------------------------------------------------------------------
    a(142): 32 ( 2
    digits)
    ------------------------------------------------------------------------------
    p(142): 29,038,737,959,536,603,197, ( 74
    digits)
    430,409,498,396,830,041,167,977,624,
    342,582,850,928,307,924,397,677,951
    ------------------------------------------------------------------------------
    q(142): 91,227,885,843,199,255,191, ( 74
    digits)
    646,869,114,045,473,262,272,556,219,
    171,168,448,644,712,866,903,622,859
    ------------------------------------------------------------------------------
    dap_h(142): 318,309,886,183,790,671,537,767,526, ( 108
    digits)
    745,028,724,068,919,291,480,912,897,
    495,334,688,117,793,595,268,474,246,
    168,599,718,714,334,570,984,041,810
    ------------------------------------------------------------------------------
    dap_k(142): 1, ( 109
    digits)
    000,000,000,000,000,000,000,000,000,
    000,000,000,000,000,000,000,000,000,
    000,000,000,000,000,000,000,000,000,
    000,000,000,000,000,000,000,000,000
    ------------------------------------------------------------------------------
    dividend(143): 17 ( 2
    digits)
    ------------------------------------------------------------------------------
    divisor(143): 7 ( 1
    digit )
    ------------------------------------------------------------------------------
    a(143): 2 ( 1
    digit )
    ------------------------------------------------------------------------------
    p(143): 58,983,398,017,341,455,772, ( 74
    digits)
    529,824,500,687,072,839,499,438,160,
    996,035,921,030,725,175,258,138,021
    ------------------------------------------------------------------------------
    q(143): 185,301,809,895,042,693,379, ( 75
    digits)
    483,012,042,516,431,758,823,861,575,
    719,915,020,219,736,793,313,904,938
    ------------------------------------------------------------------------------
    dap_h(143): 318,309,886,183,790,671,537,767,526, ( 108
    digits)
    745,028,724,068,919,291,480,912,897,
    495,334,688,117,793,595,268,474,246,
    168,599,718,714,334,570,984,041,810
    ------------------------------------------------------------------------------
    dap_k(143): 1, ( 109
    digits)
    000,000,000,000,000,000,000,000,000,
    000,000,000,000,000,000,000,000,000,
    000,000,000,000,000,000,000,000,000,
    000,000,000,000,000,000,000,000,000
    ------------------------------------------------------------------------------
    dividend(144): 7 ( 1
    digit )
    ------------------------------------------------------------------------------
    divisor(144): 3 ( 1
    digit )
    ------------------------------------------------------------------------------
    a(144): 2 ( 1
    digit )
    ------------------------------------------------------------------------------
    p(144): 147,005,533,994,219,514,742, ( 75
    digits)
    490,058,499,770,975,720,166,853,946,
    334,654,692,989,758,274,913,953,993
    ------------------------------------------------------------------------------
    q(144): 461,831,505,633,284,641,950, ( 75
    digits)
    612,893,199,078,336,779,920,279,370,
    610,998,489,084,186,453,531,432,735
    ------------------------------------------------------------------------------
    dap_h(144): 318,309,886,183,790,671,537,767,526, ( 108
    digits)
    745,028,724,068,919,291,480,912,897,
    495,334,688,117,793,595,268,474,246,
    168,599,718,714,334,570,984,041,810
    ------------------------------------------------------------------------------
    dap_k(144): 1, ( 109
    digits)
    000,000,000,000,000,000,000,000,000,
    000,000,000,000,000,000,000,000,000,
    000,000,000,000,000,000,000,000,000,
    000,000,000,000,000,000,000,000,000
    ------------------------------------------------------------------------------
    dividend(145): 3 ( 1
    digit )
    ------------------------------------------------------------------------------
    divisor(145): 1 ( 1
    digit )
    ------------------------------------------------------------------------------
    a(145): 3 ( 1
    digit )
    ------------------------------------------------------------------------------
    p(145): 500,000,000,000,000,000,000, ( 75
    digits)
    000,000,000,000,000,000,000,000,000,
    000,000,000,000,000,000,000,000,000
    ------------------------------------------------------------------------------
    q(145): 1,570,796,326,794,896,619,231, ( 76
    digits)
    321,691,639,751,442,098,584,699,687,
    552,910,487,472,296,153,908,203,143
    ------------------------------------------------------------------------------
    dap_h(145): 318,309,886,183,790,671,537,767,526, ( 108
    digits)
    745,028,724,068,919,291,480,912,897,
    495,334,688,117,793,595,268,474,246,
    168,599,718,714,334,570,984,041,810
    ------------------------------------------------------------------------------
    dap_k(145): 1, ( 109
    digits)
    000,000,000,000,000,000,000,000,000,
    000,000,000,000,000,000,000,000,000,
    000,000,000,000,000,000,000,000,000,
    000,000,000,000,000,000,000,000,000
    ------------------------------------------------------------------------------
    approx_num(-1): 4,632,228,188,615,595,682, ( 46
    digits)
    154,930,804,184,359,491,309,476,209
    ------------------------------------------------------------------------------
    approx_den(-1): 1,474,484,027,495,577,088, ( 46
    digits)
    910,665,046,416,182,563,090,448,351
    ------------------------------------------------------------------------------
    dap_num(-1): 3, ( 109
    digits)
    141,592,653,589,793,238,462,643,383,
    279,502,884,197,169,399,375,105,820,
    974,944,592,307,816,406,285,999,999,
    999,999,999,914,996,063,429,013,961
    ------------------------------------------------------------------------------
    dap_den(-1): 1, ( 109
    digits)
    000,000,000,000,000,000,000,000,000,
    000,000,000,000,000,000,000,000,000,
    000,000,000,000,000,000,000,000,000,
    000,000,000,000,000,000,000,000,000
    ------------------------------------------------------------------------------
    error_num(-1): - 62, ( 29
    digits)
    668,473,374,083,034,688,657,367,193
    ------------------------------------------------------------------------------
    error_den(-1): 737,242,013,747, ( 120
    digits)
    788,544,455,332,523,208,091,281,545,
    224,175,500,000,000,000,000,000,000,
    000,000,000,000,000,000,000,000,000,
    000,000,000,000,000,000,000,000,000
    ------------------------------------------------------------------------------
    Datesfat Chicks, May 2, 2012
    #15
  16. On 5/1/12 8:14 PM, Steve Pope wrote:
    > Tim Wescott<> wrote:
    >

    ....
    >>
    >> 245/78. It's only a bit better than twice as good as 22/7 -- then along
    >> comes 355/113, which is over 1000 times better than 245/78.

    >
    > Suppose you do the same thing with the fine structure constant --
    > let me know what you discover.
    >


    not quite m/n but

    alpha = cos(pi*137)/137 * tan(pi*(29*137))/(pi*(29*137))

    actually i think that sqrt(4*pi*alpha) = 0.30282212 is the more
    fundamental number than the fine-structure constant. the fine-structure
    constant should be thought of as a consequence of this number.

    --

    r b-j

    "Imagination is more important than knowledge."
    robert bristow-johnson, May 2, 2012
    #16
  17. On 5/1/12 7:35 PM, Joel Koltner wrote:
    > John S wrote:
    >> I like the idea that both 22 and 7 each fit into a byte whereas 355 does
    >> not. And, 22/7 is hi by only .04%. Beautiful!

    >
    > Jack Crenshaw's book, "Math Toolkit for Real-Time Programming"
    > (http://www.amazon.com/Math-Toolkit-Real-Time-Programming-ebook/dp/B003WUYQVY)
    > spends a lot of time discussing how to make "good enough" approximations
    > of various, e.g., transcendental functions... and how to know when "good
    > enough" really is. It's quite handy for this sort of thing...


    i've sent him some series that were simpler and better than his (at
    least those that were published at the time). i have no idea what rules
    of optimization he was using.

    he wrote back. didn't see anything happen about it since.

    --

    r b-j

    "Imagination is more important than knowledge."
    robert bristow-johnson, May 2, 2012
    #17
  18. John S <> writes:

    > On 5/1/2012 6:16 PM, Tim Wescott wrote:
    >> Instead of doing productive work, I just spent a few enjoyable minutes
    >> with Scilab finding approximations to pi of the form m/n.
    >>
    >> Because I'm posting to a couple of nerd groups, I can be confident that
    >> most of you probably know 22/7 off the tops of your heads.
    >>
    >> What interested me is how spotty things are -- after 22/7, the error
    >> drops for a bit until you get down to 355/113 (which, if you're at an
    >> equal level of nerdiness to me will ring a bell, but not have been
    >> swimming around in your brain to be found).
    >>
    >> But what's _really_ interesting, is that the next better fit isn't found
    >> until you get up to 52163/16604. Then things get steadily better until
    >> you hit 104348/33215 -- at which point the next lowest ratio which
    >> improves anything is 208341/66317, then 312689/99532. At this point I
    >> decided that I would post my answers for your amusement, and get back to
    >> being productive.
    >>
    >> Discrete math is so fun. And these newfangled chips are just destroying
    >> the joy, by making floating point efficient and cheap enough that you
    >> don't need to know little tricks like pi = (almost) 355/113.
    >>

    >
    > I like the idea that both 22 and 7 each fit into a byte whereas 355
    > does not. And, 22/7 is hi by only .04%. Beautiful!


    We had a teacher that insisted it was exactly equal!

    --

    John Devereux
    John Devereux, May 2, 2012
    #18
  19. Tim Wescott

    Steve Pope Guest

    John Devereux <> wrote:

    >John S <> writes:


    >> I like the idea that both 22 and 7 each fit into a byte whereas 355
    >> does not. And, 22/7 is hi by only .04%. Beautiful!

    >
    >We had a teacher that insisted it was exactly equal!


    I recall the time when you couldn't consider 25.4 mm to be exactly
    one inch. But, they fudged enough standards so that it is now exact.

    Prior to that, neither the British inch nor the American inch
    measures 25.4 ... and they deviated from that value in opposite
    direction!


    S.
    Steve Pope, May 2, 2012
    #19
  20. John Larkin <> writes:

    > On Tue, 01 May 2012 18:16:25 -0500, Tim Wescott <>
    > wrote:
    >
    >>Instead of doing productive work, I just spent a few enjoyable minutes
    >>with Scilab finding approximations to pi of the form m/n.
    >>
    >>Because I'm posting to a couple of nerd groups, I can be confident that
    >>most of you probably know 22/7 off the tops of your heads.
    >>
    >>What interested me is how spotty things are -- after 22/7, the error
    >>drops for a bit until you get down to 355/113 (which, if you're at an
    >>equal level of nerdiness to me will ring a bell, but not have been
    >>swimming around in your brain to be found).
    >>
    >>But what's _really_ interesting, is that the next better fit isn't found
    >>until you get up to 52163/16604. Then things get steadily better until
    >>you hit 104348/33215 -- at which point the next lowest ratio which
    >>improves anything is 208341/66317, then 312689/99532. At this point I
    >>decided that I would post my answers for your amusement, and get back to
    >>being productive.
    >>
    >>Discrete math is so fun. And these newfangled chips are just destroying
    >>the joy, by making floating point efficient and cheap enough that you
    >>don't need to know little tricks like pi = (almost) 355/113.

    >
    > My old HP35 calculators have a key for pi. The newer ones hide it, a
    > tiny pastel shift key thing. So I just key in 3.14. Rob down the hall
    > uses 3.
    >
    > We are increasingly using floats in embedded stuff. Our ARM LPC3250
    > has SIMD hardware FP operations.


    Even the integer-only cortex M3s we use take less than a microsecond for
    most things. On a 72MHz STM32F2:

    Double Precision:
    0.415us / 49.852 cycles /multiply
    0.378us / 45.403 cycles /add
    2.414us / 289.702 cycles /divide
    Single Precision:
    0.194us / 23.350 cycles /multiply
    0.250us / 30.052 cycles /add
    0.610us / 73.202 cycles / divide



    --

    John Devereux
    John Devereux, May 2, 2012
    #20
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