So I was watching/catching up on numberphile and watched this 74 is cracked - Numberphile . The short version is of all the numbers less than 100 they have shown that only 3 (now 2) can be expressed as the sum of 3 cubes except for a list of a few which are proven to have no solutions. I had watched the original video this one references before and while being entertained at the time, I promptly forgot about it after watching it.
I dont think many people realize or trully care that I've spent long years of my life thinking about Diophantine equations . I did it back when I was looking for new method to methodically factor primes (read 17-32 off and on) . While I came up with some interesting stuff (to me at least) I never really got anything that good. It was always a hobby/fun problem for me. The point is solving the 3 cubes problem is something the "work" i did could probably do relatively quickly if there arent tons of Residue Classes in the various modulos of 33 and 42 (the two remaining numbers). Just saying that makes me think there probably are, but if there arent merging them in to larger equations isn't very hard.
Even if there are bunches you can still get a new set of equations that dramatically cut down your search time. Getting past 10^16 is pretty remarkably easy heck getting past 10^32 might be trivial it just depends on the numbers. I think I'm gonna spend a little time on solving 33 and 42 just for fun and see how hard the problem really is. It would be neat to have my named tied to something in mathematics even if its just a simple solution to some rather mundane puzzle. And I need another hobby/would like to return to the old hobby for a while.