Problem
NT-B2-M08-P003 A Sum of Squares Modulo \(3\)
#3
★★★☆☆ Level 3 of 5
Prove that \(x^2+y^2=3z^2\) has no nonzero integer solutions.
Squares modulo \(3\) are \(0\) or \(1\).
Modulo \(3\), \(x^2+y^2\equiv 0\). This is possible only when \(x\equiv y\equiv 0\pmod 3\). Then \(9\mid x^2+y^2\), so \(9\mid 3z^2\), and \(3\mid z\). Dividing \(x,y,z\) by \(3\), we get a smaller nonzero solution. Contradiction.
Shows descent in three variables.