Problem
ALG-B1-M10-P022 Three squares modulo 8
#22
★★★★☆ Level 4 of 5
Prove that \(x^2+y^2+z^2=8k+7\) has no integer solutions.
Squares modulo \(8\) are \(0,1,4\).
Possible square residues modulo \(8\) are \(0,1,4\). A sum of three such residues is never \(7\) modulo \(8\). The right-hand side is congruent to \(7\), contradiction.
Modulo \(8\) is often stronger than modulo \(4\).