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| {{Expand French|Théorème des deux carrés de Fermat|topic=sci|date=July 2012}}
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| {{Expand Catalan|Teorema de la suma de dos quadrats|fa=yes|date=July 2013}}
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| {{For|other theorems named after Pierre de Fermat|Fermat's theorem (disambiguation){{!}}Fermat's theorem}}
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| {{see also|Pythagorean prime}}
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| In [[additive number theory]], [[Pierre de Fermat]]'s theorem on sums of two squares states that an [[Even and odd numbers|odd]] [[prime number|prime]] ''p'' is expressible as
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| :<math>p = x^2 + y^2,\,</math> | |
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| with ''x'' and ''y'' integers, [[if and only if]]
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| :<math>p \equiv 1 \pmod{4}.</math>
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| For example, the primes 5, 13, 17, 29, 37 and 41 are all congruent to 1 [[modular arithmetic|modulo]] 4, and they can be expressed as sums of two squares in the following ways:
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| :<math>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.</math>
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| 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.
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| [[Albert Girard]] was the first to make the observation (in 1632) <ref>Dickson, Ch. VI</ref> and Fermat was first to claim a proof of it.
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| 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.''
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| 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.
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| ==Proofs of Fermat's theorem on sums of two squares== | |
| {{main|Proofs of Fermat's theorem on sums of two squares}}
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| Fermat usually did not write down proofs of 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 [[Christian Goldbach|Goldbach]] on April 12, 1749. [[Joseph Louis Lagrange|Lagrange]] gave a proof in 1775 that was based on his study of [[quadratic forms]]. This proof was simplified by [[Carl Friedrich Gauss|Gauss]] in his ''[[Disquisitiones Arithmeticae]]'' (art. 182). [[Richard Dedekind|Dedekind]] gave at least two proofs based on the arithmetic of the [[Gaussian integer]]s. There is an elegant proof using [[Minkowski's theorem]] about convex sets. Simplifying an earlier short proof due to [[Roger Heath-Brown|Heath-Brown]] (who was inspired by [[Liouville]]'s idea), [[Don Zagier|Zagier]] presented a one-sentence proof of Fermat's assertion.
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| ==Related results==
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| 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 <math>p</math>:
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| *<math>p = x^2 + 2y^2 \Leftrightarrow p\equiv 1\mbox{ or }p\equiv 3\pmod{8},</math>
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| *<math>p= x^2 + 3y^2 \Leftrightarrow p\equiv 1 \pmod{3}.</math>
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| He also wrote:
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| : ''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.''
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| In other words, if ''p, q'' are of the form 20''k'' + 3 or 20''k'' + 7, then ''pq'' = ''x''<sup>2</sup> + 5''y''<sup>2</sup>. Euler later extended this to the conjecture that
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| * <math>p = x^2 + 5y^2 \Leftrightarrow p\equiv 1\mbox{ or }p\equiv 9\pmod{20},</math>
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| * <math>2p = x^2 + 5y^2 \Leftrightarrow p\equiv 3\mbox{ or }p\equiv 7\pmod{20}.</math>
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| Both Fermat's assertion and Euler's conjecture were established by Lagrange.
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| ==See also==
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| * [[Proofs of Fermat's theorem on sums of two squares]]
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| * [[Legendre's three-square theorem]]
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| * [[Lagrange's four-square theorem]]
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| ==Notes==
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| {{Reflist}}
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| ==References==
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| *[[L. E. Dickson]]. ''[[History of the Theory of Numbers]]'' Vol. 2. Chelsea Publishing Co., New York 1920
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| *Stillwell, John. Introduction to '''''Theory of Algebraic Integers''''' by Richard Dedekind. Cambridge University Library, Cambridge University Press 1996. ISBN 0-521-56518-9
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| *{{cite book | author = D. A. Cox | title = Primes of the Form x<sup>''2''</sup> + ny<sup>''2''</sup>| publisher = Wiley-Interscience | year = 1989 | isbn=0-471-50654-0 }}
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| [[Category:Additive number theory]] | |
| [[Category:Theorems in number theory]]
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| {{Link FA|ca}}
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