Vtyjkyuk

In his answer to the question Is an integer a sum of two rational squares iff it is a sum of two integer squares? André Nicolas mentions that the proof

that if $m$ is a positive integer, then $m$ is a sum of two squares if
and only if every prime of the form $4k+3$ in the prime power
factorization of $m$ occurs to an even power

is elementary but non-trivial and "goes back to Fermat. It is most easily proved by using Gaussian integers."

I wonder if in this context an "elementary proof" is a proof, that we could have found in principle in Euclid's Elements, using only methods and arguments that Euclid might have used? Or does "elementary" mean something else (but specific)?

The only reason why Euclid or one of his contemporaries did not proof the result would have been that it was to hard for them to find (but they could have stated and proved it with the methods and arguments they had at hand).

(I assume that Euclid did not proof directly that an integer is a sum of two rational squares iff it is a sum of two integer squares.)