Given that a and b are positive integers, find the conditions for which the equation a b = c has a solution.
From a = b + c, square both sides, a = b2 + 2bc + c.
|Rearranging we get,|
As the left hand side is rational, c must be rational.
Let c=x/y, where HCF(x, y)=1.
Squaring, c=x2/y2, cy2=x2.
As the left hand side divides by y2 and HCF(x2, y2)=1, the right hand side will only divide by y2 if y2=1. Hence c=x2 must be a perfect square.
Furthermore, if c is a perfect square, a = b + c will be integer, so a must also be a perfect square.