Problem
NT-B1-M03-P007 Sum of Two Squares and \(8z+7\)
#7
★★☆☆☆ Level 2 of 5
Prove that \(x^2+y^2=8z+7\) has no integer solutions.
Use squares modulo \(8\).
Squares modulo \(8\) are \(0,1,4\). A sum of two such residues can be \(0,1,2,4,5\), but not \(7\). The right side has residue \(7\). Contradiction.
Shows why modulo \(8\), not only modulo \(4\), is needed.