Fermat's theorem on sums of two squares

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

[p = x^2 + y^2,]

with x and y integers, if and only if

[p \equiv 1 \pmod.]

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.

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.

Proofs of Fermat's theorem on sums of two squares

see proofs of Fermat's theorem on sums of two squares

As was usual for claims made by Fermat, he did not provide a proof of this claim. The first proof was by Euler, who obtained a proof by infinite descent after much effort; he announced this proof in a letter to Goldbach on April 12 1749. Lagrange gave a proof in 1775, based on his study of quadratic forms, which was simplified by Gauss in his Disquisitiones Arithmeticae (art. 182). Dedekind gave at least two proofs based on the arithmetic of the Gaussian integers.

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 p\equiv 1\mboxp\equiv 3\pmod.]
• [p= x^2 + 3y^2 \Leftrightarrow p\equiv 1 \pmod.]

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 p\equiv 1\mboxp\equiv 9\pmod]
• [2p = x^2 + 5y^2 \Leftrightarrow p\equiv 3\mboxp\equiv 7\pmod]

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

References

• Stillwell, John. Introduction to Theory of Algebraic Integers by Richard Dedekind. Cambridge University Library, Cambridge University Press 1996. ISBN 0521565189

 

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