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Concept
Fermat's theorem on sums of two squares
Summary
In additive number theory, Fermat's theorem on sums of two squares states that an odd prime p can be expressed 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.
Since the Diophantus 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 generalization of Fermat's theorem is known as the sum of two squares theorem.
Albert Girard was the first to make the observation, characterizing the positive integers (not necessarily primes) that are expressible as the sum of two squares of positive integers; this was published in 1625. The statement that every prime p of the form 4n+1 is the sum of two squares is sometimes called Girard's theorem. For his part, Fermat wrote an elaborate version of the statement (in which he also gave the number of possible expressions of the powers of p as a sum of two squares) in a letter to Marin Mersenne dated December 25, 1640: for this reason this version of the theorem is sometimes called Fermat's Christmas theorem.
Fermat's theorem on sums of two squares is strongly related with the theory of Gaussian primes.
A Gaussian integer is a complex number such that a and b are integers. The norm of a Gaussian integer is an integer equal to the square of the absolute value of the Gaussian integer.
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