Problem
NT-B1-M06-P023 Another Discriminant
#23
★★★★☆ Level 4 of 5
Prove that \(x^2+2y^2=5xy\) has no positive integer solutions.
The discriminant in \(x\) must be a square.
Rewrite as \(x^2-5yx+2y^2=0\). The discriminant \(D=25y^2-8y^2=17y^2\) must be an integer square. This would mean \(u^2=17y^2\) for some positive \(u\). But \(u^2=17y^2\) is impossible by the same descent with prime \(17\). Contradiction.
Checks whether the student understands the general scheme \(u^2=py^2\).