Problem
NT-B2-M02-P007 Residue Seven
#7
★★★☆☆ Level 3 of 5
Prove that \(x^2+y^2=8z+7\) has no integer solutions.
Squares modulo \(8\) are \(0,1,4\).
A sum of two elements from \(\{0,1,4\}\) modulo \(8\) can be \(0,1,2,4,5\). Residue \(7\) is impossible. The right-hand side is \(7\pmod8\), so there are no solutions.
Here it is important not to say that a sum of squares cannot be \(7\) without checking all residues.