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Prove that if n is the product of two numbers that can be written as the sum of two squares then n can be written as the sum of two squares.

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See the Brahmagupta–Fibonacci identity.

Or consider the formula for the product of two complex numbers, and take the moduli.

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A number can be written as the sum of two squares iff each of its prime divisors of the form $p = 4k + 3$ occurs to an even power. So if $n = ab$ and $a$ and $b$ have this property, so will their product.

See Integer as sum of Two Squares

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    $\begingroup$ This is quite an heavy machinery for such a question ! ;) $\endgroup$ – jibe Jun 20 '14 at 13:02

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