Fermat's Theorem On Sums of Two Squares

Fermat's Theorem On Sums Of Two Squares

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

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.

Albert Girard was the first to make the observation (in 1632) and Fermat was first to claim a proof of it. Fermat 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 that 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 of n's odd prime factors congruent to 3 modulo 4 occur to an even exponent, it is expressible as a sum of two squares. The converse also holds.