Sums of two squares

Fermat's theorem on sums of two squares

In number theory, Pierre de Fermat's theorem on sums of two squares states that an odd prime p is expressible as

p = x^2 + y^2,,

with x and y integers, if and only if

p equiv 1 pmod{4}.

The theorem is also known as Thue's Lemma, after Axel Thue.

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:

5 = 1^2 + 2^2, quad 13 = 2^2 + 3^2, quad 17 = 1^2 + 4^2, quad 29 = 2^2 + 5^2, quad 37 = 1^2 + 6^2, quad 41 = 4^2 + 5^2.

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.

According to Ivan M. Niven, Albert Girard was the first to make the observation 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, this shows that any positive integer, all of whose odd prime factors congruent to 3 modulo 4 occur to an even exponent, is expressible as a sum of two squares. The converse also holds.

Proofs of Fermat's theorem on sums of two squares

Fermat usually did not prove his claims and he did not provide a proof of this statement. The first proof was found by Euler after much effort and is based on infinite descent. He announced it in a letter to Goldbach on April 12, 1749. Lagrange gave a proof in 1775 that was based on his study of quadratic forms. This proof was simplified by Gauss in his Disquisitiones Arithmeticae (art. 182). Dedekind gave at least two proofs based on the arithmetic of the Gaussian integers. There is an elegant proof using Minkowski's theorem about convex sets. Simplifying an earlier short proof due to Heath-Brown (who was inspired by Liouville's idea), Zagier presented a one-sentence proof of Fermat's assertion.

Related results

Fermat announced two related results fourteen years later. In a letter to Blaise Pascal dated September 25, 1654 he announced the following two results for odd primes p:

  • p = x^2 + 2y^2 Leftrightarrow pequiv 1mbox{ or }pequiv 3pmod{8}.
  • p= x^2 + 3y^2 Leftrightarrow pequiv 1 pmod{3}.

He also wrote:

If two primes which end in 3 or 7 and surpass by 3 a multiple of 4 are multiplied, then their product will be composed of a square and the quintuple of another square.

In other words, if p, q are of the form 20k + 3 or 20k + 7, then pq = x2 + 5y2. Euler later extended this to the conjecture that

  • p = x^2 + 5y^2 Leftrightarrow pequiv 1mbox{ or }pequiv 9pmod{20},
  • 2p = x^2 + 5y^2 Leftrightarrow pequiv 3mbox{ or }pequiv 7pmod{20}.

Both Fermat's assertion and Euler's conjecture were established by Lagrange.


  • Stillwell, John. Introduction to Theory of Algebraic Integers by Richard Dedekind. Cambridge University Library, Cambridge University Press 1996. ISBN 0-521-56518-9
  • D. A. Cox (1989). Primes of the Form x2+ny2. Wiley-Interscience. ISBN 0-471-50654-0.

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