Problem
NT-B2-M08-P001 The Equation \(x^2=2y^2\)
#1
★★☆☆☆ Level 2 of 5
Prove that \(x^2=2y^2\) has no solutions in positive integers.
Show that a solution forces both numbers to be even.
If \(x^2=2y^2\), then \(x^2\) is even, so \(x=2u\). Then \(4u^2=2y^2\), hence \(y^2=2u^2\), and \(y\) is also even: \(y=2v\). We get \(u^2=2v^2\), a smaller positive solution. Infinite descent is impossible.
A basic problem on the mechanism of descent.