What it boils down to is: Are there infinitely many 4*n! + 1 which only have factors of shape 4k + 1 (or possibly some factors 4k + 3, but to an even power). I'm inclined to think ``yes'', but this is far beyond our reach. It may even be true that there are infinitely many primes 4*n! + 1 ???? R. On Fri, 10 Sep 2010, Dan Asimov wrote:
Very interesting, Richard -- and Ed !
This suggests a few questions to me, possible hard ones:
1) Can we prove there are infinitely many solutions to
T[x] + T[y] = z! ? How about a probabilistic heuristic?
2) Generally, given reasonably simple functions
F,G: Z+ -> Z+
when can we prove there are infinitely many solutions to
F(x) + F(y) = G(z) ?
Likewise, what about a probabilistic "proof" ?
--Dan ----------------------------------------------
Richard wrote:
<< . . . Solutions of
x(x+1)/2 + y(y+1)/2 = z!
are solutions of (2x+1)^2 + (2y+1)^2 = 8(z!) + 2 and the first few values of z for which there are solutions can easily be ascertained:
(x,y,z) = (0,1,0), (0,1,1), (1,1,2), (0,3,3), (2,2,3), (2,6,4), (0,15,5), (5,14,5), (45,89,7), (89,269,8), (210,825,9), (760,2610,10), (1770,2030,10), . . .
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