Problem
NT-B1-M06-P010 A Sum of Squares Modulo \(3\)
#10
★★☆☆☆ Level 2 of 5
Prove that if \(3\mid x^2+y^2\), then \(3\mid x\) and \(3\mid y\).
A square modulo \(3\) is \(0\) or \(1\).
Squares modulo \(3\) are only \(0\) and \(1\). For the sum of two such residues to be divisible by \(3\), the only possibility is \(0+0\). Thus \(x^2\equiv0\pmod3\) and \(y^2\equiv0\pmod3\). Therefore \(3\mid x\) and \(3\mid y\).
This is the lemma for descent in \(x^2+y^2=3z^2\).