Problem
NT-B1-M05-P012 Obstruction Modulo \(3\)
#12
★★☆☆☆ Level 2 of 5
Prove that the equation \(x^2=3y^2+2\) has no integer solutions.
Consider the equation modulo \(3\).
Modulo \(3\), we get \(x^2\equiv2\pmod3\). But a square modulo \(3\) can have only residue \(0\) or \(1\). This is a contradiction, so no solutions exist.
A short problem on an impossible quadratic residue.