The pairs of positive integers (A, B) are such that B divides 2A².

Prove that A² + B cannot be a perfect square.

Given (A,B) are positive integers and B is a divisor of
2AČ.
So, B = 2AČ/k where k is also a positive integer
AČ+B=AČ+2AČ/k=AČ(k+2)/k
There dont two perfect squares with a difference of 2 or less
(in case k is even)
√((k+2)/k) can't be perfect rational number.
So, AČ+B can't be perfect square.

Posted by Praneeth on 2008-05-09 04:00:07