# If $n=x^2+3y^2$ then any prime in $n$'s factorization is of an even power.

If $n=x^2+3y^2$ then any prime $p$ such that $p\equiv 2 \pmod 3$ in $n$'s factorization is of an even power. I have been spending hours trying to solve this because of some some issues withholding any progress when I don't understand something. I came be the other duplicate of this question and didn't understand a thing either. I could really use your help.

(Major) hint: Note that the product of an even number of primes $\equiv 2\pmod{3}$ is $\equiv 1\pmod{3}$, while the product of an odd number is $\equiv 2\pmod{3}$. But the right-hand side is $x^2+3y^2\equiv x^2\pmod{3}$, which cannot be $\equiv 2\pmod{3}$.