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I need help with the following two generalities:

What is the characterization of primes that are sums of two squares?

And what is the characterization of $n\ge 1$ that are sums of two squares?

I'm hoping I'm over thinking these two generalities, but they seem extremely broad to me.

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    Take a look [here](http://en.wikipedia.org/wiki/Fermat's_theorem_on_sums_of_two_squares).2012-11-09

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This is sometimes called Fermat's Christmas theorem. The primes are $2$ and all those of the form $4k+1$. For composites, all prime factors of the form $4k+3$ must occur an even number of times.