A positive integer is called a *squaresum* if and only if it can be written as the sum of the squares of two integers. For example, $61$ and $9$ are both squaresums since $61=5^2+6^2$ and $9=3^2+0^2$.

A prime number is called *awkward* if and only if it has a remainder of $3$ when divided by $4$. For example, $23$ is awkward since $23=5\times 4+3$.

A (true) theorem due to Fermat states that:

A positive integer is a squaresum if and only if each of its awkward prime factors occurs to an even power in its prime factorisation.

It follows that $5\times 23^2$ is a squaresum, since $23$ occurs to the power $2$, but $5\times 23^3$ is not, since $23$ occurs to the power $3$.

Which one of the following statements is not true?