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.

It looks like **yes**, n=56 creates 169 and 225 ,both perfect squares.

Evaluating members manually of two series a1=3*m*56+1 and a2=4*m*56+1 ,I did not find any m number causing a perfect square number in both series.

That is not a proof ,but unless a counterexample is found I

believe that N=56 might be the only case and therefore the following statvement is true:

**If** **N is a positive integer such that each of 3*N + 1 and 4*N + 1 is a perfect square-than N is divisible by 56.**

...still waiting for more cases or counterexample...

*Edited on ***April 2, 2010, 4:36 am**