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.

I ran a brute force search up to 500,000,000 and found four values:
56=56*1,
10920=56*195
2118480=56*37830
410974256=56*7338826

These values are approximated by 56*10^(2.2878*k+.0022) for k=0 to 3.  Taking k=4 suggests the next solution to be in the neighborhood of 56*10^9.1534=79,723,000,000

Edit: My test just finished, fifth value confirmed one answer (no non-multiples of 56 showed up): 79,762,887,240=56*1,423,694,415

Edited on April 1, 2010, 8:59 pm

Posted by Brian Smith on 2010-04-01 20:55:46