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.

Testing N up to a million, the only two cases where the factors (i.e. 3N+1 and 4N+1) are both perfect squares did yield divisibility by 56:

56 * 1 --> 13*13=169 and 15*15=225

56 * 195 --> 181*181= 32761 and 209*209 = 43681

Not a proof -- should I suspect something since this is posed on April Fools Day?