ALG-B1-M11-P019 Descent

Choose a minimal solution and replace the larger root by the other root of the quadratic equation.

Assume a solution exists, \(x\ge y\), with minimal \(x+y\). The case \(x=y\) is impossible. Consider \(X^2-7yX+y^2=0\). The other root \(x'=7y-x=\frac{y^2}{x}\) is positive, integral, and smaller than \(y\). Then \((x',y)\) is a smaller positive solution. Contradiction.

Strategy: Vieta descent as a non-obvious method choice.