## Impossible Solution

#### Problem

Given that `a` and `b` are positive integers, find the conditions for which the equation `a` `b` = `c` has a solution.

#### Solution

From `a` = `b` + `c`, square both sides, `a` = `b`^{2} + 2`b``c` + `c`.

Rearranging we get, | a b^{2} c2 b | = c. |

As the left hand side is rational, `c` must be rational.

Let `c`=`x`/`y`, where HCF(`x`, `y`)=1.

Squaring, `c`=`x`^{2}/`y`^{2}, `cy`^{2}=`x`^{2}.

As the left hand side divides by y^{2} and HCF(`x`^{2}, `y`^{2})=1, the right hand side will only divide by `y`^{2} if `y`^{2}=1. Hence `c`=`x`^{2} 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.

Problem ID: 190 (28 Nov 2004) Difficulty: 3 Star