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 2010-04-03 21:23:59