Problem
NT-B1-M05-P018 A Sum of Squares Modulo \(8\)
#18
★★★☆☆ Level 3 of 5
Prove that the equation \(x^2+y^2=8z+6\) has no integer solutions.
List the possible residues of a square modulo \(8\).
A square modulo \(8\) is \(0,1\), or \(4\). A sum of two such residues can be \(0,1,2,4,5\) modulo \(8\), but not \(6\). The right-hand side has residue \(6\), a contradiction.
Repeats the example idea in an independent formulation.