Question

Find a counter example to the statement that every even positive integer can be written as the sum of the squares of three integers.

Answer 1

Consider the integer number $n=28$. Let us show that $n$ can not be written as the sum of the squares of three integers. For this prove that the equation $a+b+c=28$ has no solution, where $a,b,c$ are squares of integers.

It followst that $a\geq0, b\geq 0, c\geq 0$, and therefore $a\leq 28, b\leq 28, c\leq 28$. Taking into account that $a,b,c$ are squares of integers, we conclude that $a,b,c\in X=\{0,1,4,9,16,25\}$.

Suppose that $a=25$. Then $b+c=3$, and we see that there are no $b,c\in X$ such that $b+c=3$. By analogy for the cases $b=25$ and $c=25$.

Suppose that $a=16$. Then $b+c=12$, and we see that there are no $b,c\in X$ such that $b+c=12$. By analogy for $b=16$ and $c=16$.

If $a\leq 9, b\leq 9, c\leq 9$, then $a+b+c\leq 27$, and there are no $a,b,c\in X$ such that $a+b+c=28$.

Therefore, $n=28$ can not be written as the sum of the squares of three integers.