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.

 

1.       56 = 7*8. The part of the solution showing 8 is a factor has already been done.<o:p></o:p>

2.       For all m, m^2-1 = (m-1)(m+1)<o:p></o:p>

3.       3n+1 is a perfect square, x ^2. 3n= (x-1)(x+1) = k<o:p></o:p>

4.       4n+1 is a perfect square, y^2.  4n= (y-1)(y+1) = l<o:p></o:p>

5.       7n = k+l<o:p></o:p>

6.       n=(k+l)/7<o:p></o:p>

7.       n is an integer. Therefore n is divisible by 7.<o:p></o:p>

 

Posted by broll on 2010-04-04 04:09:42