Are there no even squares expressible as the sum of two prime squares?

Question

When I was playing around with different number sequences, I noticed that I couldn't find any even squares that are expressible as the sum of two prime squares. Is this true, and is this related to Fermat's theorem on sums of two squares?

Answer 1

Hint: Even squares are of necessity multiples of four $($why ?$)$. The sum of two odd prime squares is of necessity of the form $4k+2~($why ?$)$.

Answer 2

Yes, it's true, but I don't think there's a direct relevance between Fermat's theorem on sums of two squares what you're trying to prove.

For what you're trying to prove, it's enough to look at squares modulo 4. If $n \equiv 2 \pmod 4$, then $n^2 \equiv 0 \pmod 4$. If $p \equiv 3 \pmod 4$, then $p^2 \equiv 1 \mod 4$. So it's really easy to see that $1 + 1 \not\equiv 0 \pmod 4$.

What Fermat was looking at was which primes are the sum of two squares. Examination of small primes suggests each prime $p$ has such a representation as long as $p \not\equiv 3 \pmod 4$. Proving this is not as difficult as proving Fermat's favorite conjecture, but it is a little more involved than your question about even squares. A paper by Pete Clark might be good reading if you're interested in exploring sums of two squares further.