examday

/TMUA

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 $23=5\times4+3$.

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 $5\times23^2$ is a squaresum, since 23 occurs to the power 2, but $5\times23^3$ is not, since 23 occurs to the power 3.

Which one of the following statements is not true?