Studio 1558 CPU upgrade issue

Two beeps means no memory is being recognized.  Have you tried reinstalling the memory module(s)?  If the problem doesn't occur with the i3 but does with the i5, you've likely a bad memory controller - which is part of the CPU.  

Thank you for your reply. I tried with 3 pairs of ram, all of which work with the i3 but have the same beep problem with the i5 - i am sorry, I meant I get 4 beeps, not 2. I tested to see if I get beeps without the ram, and that's when I get 2 beeps, which is expected. When I tried with just 1 stick of ram (on slot 1) with the i5, the laptop boots to a black screen and stays that way, no POST, no beeps, no nothing, but the fan is working. Any ideas?

Yes, the board for the i7 is different for sure, it's a quad-core CPU. The labels on the i3 board are pretty much the same with the i5 version of the laptop, so I was wondering whether it's some strange dell micro-revision limitation on upgrades...

So if I decided to buy another i5 CPU - i have seen versions of this laptop with i5-520m, i5-540m, i5-560m - do you think they would work (does CPU stepping count)? I know it's an old laptop but I really love this machine and I am planning to buy a new laptop when Coffee Lake laptops become mainstream.

I don't think so.  It's true that systems of that age sometimes use a different board revision for i7 CPUs, but the i3 and i5 have just about identical board requirements.

I see. And is it possible that there is a motherboard revision of the 1558 that supports only the i3 but not the i5 processors?

I would venture that means the i5 CPU you have, has a damaged memory controller.  Both those CPUs use the same RAM - and if the board works with all of the modules and the i3, but not the i5 -- the i5 CPU is somehow damaged (could be internal, or could be damage to one or more pins on the CPU).

The i5-540m was shipped with this model - whether there are differences in board revision may be difficult or impossible to determine.  These were first generation Core CPUs -- there were revisions to them and to boards along the way even more so than usual.