Problem
NT-B1-M06-P017 A Primitive Solution Is Impossible
#17
★★★☆☆ Level 3 of 5
Prove that there are no coprime positive integers \(x,y\) such that \(x^2=2y^2\).
Show that both numbers must be even.
From \(x^2=2y^2\), \(x\) is even, \(x=2u\). Then \(y^2=2u^2\), so \(y\) is even. Thus \(x\) and \(y\) have common divisor \(2\), contradicting coprimality.
This version avoids repetition: contradiction with primitivity.