Sony Vaio not booting up at all

I have a strange problem with a vaio vgn-s1hp

Sporadically when turned on it won't boot up. No splash screen, all
you get is the LED's on the front. There is no hard disc activity or
any processing noise whatsoever.

Sounds like damaged memory or memory slots?

I would agree, but as soon as you plug in a monitor and boot it up it
is fine on both laptop and monitor screen, even if you remove the
monitor from the VGA, it works fine and then will work fine afterwards
as well. It may just be coincidence, but it is as if the monitor wakes
it up. It is not he screen sharing or else you would get some noise
from the laptop and possibly the splash screen.

Help please.