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 youíve moved us on somewhat, but Iím 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.
Iím beginning to think that the squared numbers themselves are the key to solving this problem. Iíll try to post something tomorrow.
Posted by Harry
on 2010-04-03 21:23:59