Question

Find a counterexample to the statement “every positive integer greater than 7 can be written as the sum of the squares of three (not necessarily unique) integers.”

Answer 1

Legendre's three-square theorem states that a natural number can be represented as the sum of three squares of integers ${\displaystyle n=x^{2}+y^{2}+z^{2}}$ if and only if $n$ is not of the form ${\displaystyle n=4^{a}(8b+7)}$  for nonnegative integers $a$ and $b.$

The first numbers that cannot be expressed as the sum of three squares (i.e. numbers that can be expressed as ${\displaystyle n=4^{a}(8b+7)})$ are

$${\displaystyle n=4^{0}(8(0)+7)}=7$$

$${\displaystyle n=4^{0}(8(1)+7)}=15>7$$

$${\displaystyle n=4^{0}(8(2)+7)}=23$$

$${\displaystyle n=4^{1}(8(0)+7)}=28$$

$${\displaystyle n=4^{0}(8(2)+7)}=23$$

$${\displaystyle n=4^{0}(8(3)+7)}=31$$

$${\displaystyle n=4^{0}(8(4)+7)}=39$$

$${\displaystyle n=4^{0}(8(5)+7)}=47$$

$${\displaystyle n=4^{0}(8(6)+7)}=55$$

$${\displaystyle n=4^{1}(8(0)+7)}=60$$

$$...$$