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I'd really like your help with the following Number Theory question:

I need to show that if I can write an integer $n=x^2+3y^2$ so in the factorization of $n$ to primes, every $p \equiv 2\pmod 3$ would be with a even power, I mean if $n=\prod p^{a(p)}$ so $a(p)$ is even for all $p \equiv 2 \pmod 3$, where $p$ is prime.

I don't rally know how to start this one.

Thank you.

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Hint: If $p\neq 3$ is a prime and $p|x^2+3y^2$ and $p\not\mid x$, $p\not\mid y$, then $-3$ is a square mod $p$.

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    You can think of this as a proof by descent. Assume there is a least $n=x^2+3y^2$ which is a counter-example. Then there is some $p\equiv 2\pmod 3$ which goes into $n$ an odd number of times. But then, by the above, $p|x$ and $p|y$, so $n/p^2 = (x/p)^2+3(y/p)^2$ is a smaller counter-example.2012-07-05