Problem
NT-B1-M11-P017 Sum of Two Squares
#17
★★★☆☆ Level 3 of 5
Prove that a number of the form \(4k+3\) cannot be represented as a sum of two integer squares.
Use square residues modulo \(4\).
A square modulo \(4\) is \(0\) or \(1\). A sum of two squares modulo \(4\) can only be \(0,1,2\). Residue \(3\) is impossible. Hence \(4k+3\) is not a sum of two squares.
Short but fundamental lemma.