Since the Pentium 4's were produced from 2000-2008 (according to Wikipedia), some of them should be able to work with 64bit OS. Think it was around 2006 when many processors were x64 from there on (which means x86-64 so they work with both x86 (32bit) and x64 (64bit)).
The biggest problem is that a Pentium 4 isn't a very powerful processor. I own many of them from 2000 and onward. I ran Windows XP great on them, Windows 7 was also ok but the integrated graphics will not support aero. Aero should work fine with your GPU (I personally own a Gigabyte Geforce 210 1GB edition - so I know is can run any OS. No matter how crazy the effects are (including the crap that Linux has - that real crazy wobbly windows nonsense that looks dumb. lol) Of course these things take a ride on the CPU as well and Pentium 4 does not handle them well.
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@safe1st you will experience some performance problems with that CPU and while official specs say 4GB, I personally recommend 5GB and up. In general, while you will lose 512GB of RAM using a 32bit OS with 4GB RAM (32bit only supports 3.5GB of RAM), you will have much better performance.
Windows 7 32bit (any edition) should work fine. I've never had a key for Windows 7, 8(.1), or 10 that didn't accept both 32 and 64bit versions of the edition. For example, Windows 7 Professional 32bit key will work on a 64bit edition of Windows 7 Professional.