Problem
NT-B2-M08-P004 A Sum of Squares Modulo \(7\)
#4
★★★☆☆ Level 3 of 5
Prove that \(x^2+y^2=7z^2\) has no nonzero integer solutions.
The squares modulo \(7\) are \(0,1,2,4\). Which two of them can sum to \(0\)?
Squares modulo \(7\) are \(0,1,2,4\). A sum of two such residues can be \(0\) modulo \(7\) only if both residues are \(0\). Hence \(7\mid x\) and \(7\mid y\). Then \(49\mid x^2+y^2\), so \(7\mid z\). Dividing all variables by \(7\) gives a smaller solution, impossible.
A good exercise in building a square-residue table.