N is a positive integer such that each of 3*N + 1 and 4*N + 1 is a perfect square.
Is N always divisible by 56?
If so, prove it. Otherwise, give a counterexample.
(In reply to
Modular arithmetic by Brian Smith)
Well done Brian, I think you’ve moved us on somewhat, but I’m still not sure how this modulo 7 approach can be completed.
Incidentally, I notice that in your 4th paragraph you claim that N = 7x + 4 remains a possibility, but I think that should be N = 7x + 5, which then gives
4N + 1 = 28x + 21 rather than 4N + 1 = 28x + 17.
This means that both N = 8(7x + 2) and N = 8(7x + 5) still need to be ruled out to prove that N is a multiple of 56.
I’m beginning to think that the squared numbers themselves are the key to solving this problem. I’ll try to post something tomorrow.

Posted by Harry
on 20100403 21:23:59 