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

Suppose a^2+b^2/a^2-b^2 =k (an integer)

Then it follows that a^2/b^2 = k+1/k-1...the LHS is in its lowest reducible form (cos gcd(k-1,k+1)=1)

This implies k+1 and k-1 are each squares of some positive integers...But this isnt possible bcos the minimum difference btwn 2 squares is 3 (but here it is 2).

Thus our assumption that k is an integer isnt true.

Thus proved

Posted by lpriya on 2007-10-16 07:50:59