If a and b are distinct positive integers, then show that (a²+b²)/(a²-b²) can not be an integer.

The expression equals 1 + (2b^2)/(a+b)(a-b).

(a+b) divides 2b^2 evenly only if
a = 0 or b or (b^2 - b) or (2b^2 - b).

The problem condition rules out a = 0 or a = b.

If a = (b^2 - b), then the expression =
   1 + 2/(b^2 - 2b), which is an integer only if b = 1.
   but then a = 0, which has been ruled out.

If a = (2b^2 - b), then the expression =
   1 + 1/(b^2 - 2b), which is never an integer.

q.e.d.

Posted by Steve Herman on 2007-10-07 22:44:25