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In additive number theory, Pierre de Fermat's theorem on sums of two squares states that an odd prime p is expressible as with x and y integers, if and only if The prime numbers for which this is true are called Pythagorean primes.For example, the primes 5, 13, 17, 29, 37 and 41 are all congruent to 1 modulo 4, and they can be expressed as sums of two squares in the following ways:

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• Fermat's theorem on sums of two squares
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• In additive number theory, Pierre de Fermat's theorem on sums of two squares states that an odd prime p is expressible as with x and y integers, if and only if The prime numbers for which this is true are called Pythagorean primes.For example, the primes 5, 13, 17, 29, 37 and 41 are all congruent to 1 modulo 4, and they can be expressed as sums of two squares in the following ways:
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• In additive number theory, Pierre de Fermat's theorem on sums of two squares states that an odd prime p is expressible as with x and y integers, if and only if The prime numbers for which this is true are called Pythagorean primes.For example, the primes 5, 13, 17, 29, 37 and 41 are all congruent to 1 modulo 4, and they can be expressed as sums of two squares in the following ways: On the other hand, the primes 3, 7, 11, 19, 23 and 31 are all congruent to 3 modulo 4, and none of them can be expressed as the sum of two squares. This is the easier part of the theorem, and follows immediately from the observation that all squares are congruent to 0 or 1 modulo 4. Albert Girard was the first to make the observation, describing all positive integral numbers (not necessarily primes) expressible as the sum of two squares of positive integers; this was published posthumously in 1634. Fermat was the first to claim a proof of it; he announced this theorem in a letter to Marin Mersenne dated December 25, 1640: for this reason this theorem is sometimes called Fermat's Christmas Theorem. Since the Brahmagupta–Fibonacci identity implies that the product of two integers each of which can be written as the sum of two squares is itself expressible as the sum of two squares, by applying Fermat's theorem to the prime factorization of any positive integer n, we see that if all the prime factors of n congruent to 3 modulo 4 occur to an even exponent, then n is expressible as a sum of two squares. The converse also holds. This equivalence provides the characterization Girard guessed.
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