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

Writing the expression in the problem as a mixed number gives 1+(2b^2/(a^2-b^2)). If a and b exist that aren't relatively prime, then there exists a case as well where a and b are relatively prime by dividing by the common factor.

Take the integer the expression is to be 1+k. Then it follows that 1+k=1+(2b^2/(a^2-b^2)), k(a^2-b^2)=2b^2, ka^2=(k+2)b^2

This results in k/(k+2)=b^2/a^2, and then assuming k is an integer leads to a contradiction since k/k+2 can't be a perfect square given the restrictions on a and b. So the expression can't be an integer.

Posted by Gamer on 2007-10-08 20:30:45