Problem
NT-B2-M08-P002 The Equation \(x^2=5y^2\)
#2
★★☆☆☆ Level 2 of 5
Prove that \(x^2=5y^2\) has no solutions in positive integers.
If \(5\mid x^2\), then \(5\mid x\).
From \(x^2=5y^2\), \(5\mid x^2\), hence \(x=5u\). Then \(25u^2=5y^2\), so \(y^2=5u^2\), and \(5\mid y\). With \(y=5v\), we get \(u^2=5v^2\), a smaller solution. Impossible.
Useful as a transition from parity to a prime divisor.