Problem
NT-B1-M03-P024 Infinitely Many Numbers Are Not Sums of Three Squares
#24
★★★★★ Level 5 of 5
Prove that infinitely many positive integers cannot be represented as a sum of three integer squares.
Consider numbers of the form \(8t+7\).
From the problem on three squares modulo \(8\), a sum of three squares cannot have residue \(7\) modulo \(8\). Therefore no number of the form \(8t+7\) can be represented as \(x^2+y^2+z^2\). There are infinitely many such positive integers: \(7,15,23,31,\ldots\). Hence infinitely many required numbers exist.
Level 5 for Book 1: one must not only get a contradiction, but construct an infinite family.