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

Hi!

From the (a²+b²)/(a²-b²) = k with k>1 positive integer we find

a²(k-1) = b²(k+1).

Multiply with (k-1)

a²(k-1)² = b²(k²-1) and a(k-1) = bsqrt(k²-1)

So sqrt(k²-1) = a(k-1)/b but ........

Square root from an integer can't be a fraction only if is a posivite integer.

But k²-1 can't be a square because (k-1)²<k²-1<k²

 

 

Posted by Chesca Ciprian on 2007-10-09 10:40:08