My uncle's factorization algorithms
This is very interesting. Thank you for devoting your time to putting this on the web. Otherwise it might have rotted in your attic!It's unclear to me whether there is anything theoretically new in here yet. The algorithms are all familiar to me except for the  P^2 + 1 and P^2 + P + 1 algorithms. But don't read too much into that. They are probably well known.What I can tell you is that there is an extraordinary amount of work in implementing all those algorithms. I do this stuff for a living and it is a tremendously impressive feat for a single individual over any span of time.I'd be interested in hearing how fast the MPQS is. The state of the art for factoring a number like 840931001586212064794450601167289569811131781103613687750579 on a single core is probably around 6s or so on a modern 2GHz x86 processor.UBASIC already had a 32 bit x86 assembly optimised MPQS in it, and it performed pretty well actually. So it would be interesting if your uncle improved on that, especially if he had theoretical improvements.The state of the art for the number field sieve should factor the 79 digit number here: http://www.loria.fr/~zimmerma/records/rsa.html in around 10-20 minutes on a single core, though in general the GNFS is for much larger integers. It seems unlikely that a little UBASIC program could manage those, however.Having said that, I am still in shock that your uncle actually implemented the GNFS. That is a staggering accomplishment and is something to be genuinely impressed by.I don't know if you have a gold mine there or the cutting edge life's work of someone from yesteryear.