Memtest checks if there are any problems with your RAM. I am not aware of a way to run it in vector. The way I do it is I have a Ubuntu install CD, and once I boot from it it gives an option to run a memtest. You'll need to leave your laptop running memtest overnight for it to produce any meaningful results. If by then it gives no errors, then your RAM is probably alright.
It doesn't seem like a RAM problem to me, just thought it's worth to make sure.
What it does look like is a video card related problem(given the screen going black and all).
It seems both of your laptops have an Intel GMA 950 video card.
I personally have no experience with intel graphics cards, but i think it's worth looking into it, as that could be the source of the problem.
I can see you're using "intel" graphics driver on this laptop. What about the other one?
Again, I am no expert, so it could be something totally different.