Problem
NT-B1-M03-P013 Three Squares and \(8t+7\)
#13
★★★☆☆ Level 3 of 5
Prove that a number of the form \(8t+7\) cannot be represented as a sum of three integer squares.
Squares modulo \(8\) are \(0,1,4\). Check sums of three such residues.
A square modulo \(8\) is \(0,1\), or \(4\). A sum of three such residues can give \(0,1,2,3,4,5,6\), but not \(7\). Hence a sum of three squares cannot be congruent to \(7\) modulo \(8\). The number \(8t+7\) has residue \(7\). Contradiction.
This is a strong classical template, but the solution remains accessible.