In mathematics, Pierre de Fermat's theorem on sums of two squares states that an odd prime number p is expressible as Euclid, detail from The School of Athens by Raphael. ... Pierre de Fermat Pierre de Fermat (August 17, 1601 – January 12, 1665) was a French lawyer at the Parliament of Toulouse and a mathematician who is given credit for the development of modern calculus. ... In mathematics, any integer (whole number) is either even or odd. ... In mathematics, a prime number (or a prime) is a natural number that has exactly two (distinct) natural number divisors, which are 1 and the prime number itself. ...

p = x² + y²,

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: Modular arithmetic (sometimes called modulo arithmetic) is a system of arithmetic for integers, where numbers wrap around after they reach a certain value — the modulus. ...

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. Marin Mersenne, Marin Mersennus or le Père Mersenne (September 8, 1588 – September 1, 1648) was a French theologian, philosopher, mathematician and music theorist. ... December 25 is the 359th day of the year (360th in leap years) in the Gregorian Calendar, with 6 days remaining. ... Events December 1 - Portugal regains its independence from Spain and João IV of Portugal becomes king. ...

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. Fermats theorem on sums of two squares states that an odd prime can be expressed as with and integers if and only if The only if clause is trivial: the squares modulo are and , so is congruent to , , or modulo . ... Leonhard Euler aged 49 (oil painting by Emanuel Handmann, 1756) Leonhard Euler (April 15, 1707 - September 18, 1783) (pronounced oiler) was a Swiss mathematician and physicist. ... Christian Goldbach (March 18, 1690 - November 20, 1764), was a Prussian mathematician, who was born in Königsberg, Prussia, as son of a pastor. ... April 12 is the 102nd day of the year in the Gregorian calendar (103rd in leap years). ... Events While in debtors prison, John Cleland writes Fanny Hill (Memoirs of a Woman of Pleasure). ... Lagrange may mean: Joseph Louis Lagrange, mathematician and mathematical physicist A Lagrange point in physics and astronomy The Lagrange_Multiplier mathematical technique Places in the United States: Lagrange, Georgia Lagrange, Indiana Lagrange, Maine Lagrange, New York (three places): Lagrange, Dutchess County Lagrange, Orange County Lagrange, Wyoming County Lagrange, Ohio Lagrange, Virginia... 1775 was a common year starting on Sunday (see link for calendar). ... In mathematics, a quadratic form is a homogeneous polynomial of degree two in a number of variables. ... The gauss, abbreviated as G, is the cgs unit of magnetic flux density or magnetic induction (B), named after the German mathematician and physicist Carl Friedrich Gauss. ... The Disquisitiones Arithmeticae is a textbook of number theory written by German mathematician Carl Friedrich Gauss and first published in 1801 when Gauss was 24. ... Julius Wilhelm Richard Dedekind (October 6, 1831 - February 12, 1916) was a German mathematician and Ernst Eduard Kummers closest follower in arithmetic. ... A Gaussian integer is a complex number whose real and imaginary part are both 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: Blaise Pascal (June 19, 1623–August 19, 1662) was a French mathematician, physicist, and religious philosopher. ... September 25 is the 268th day of the year (269th in leap years). ... Events April 5 - Signing of the Treaty of Westminster, ending the First Anglo-Dutch War. ...

• or
• 

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

• or
• or

Both Fermat's assertion and Euler's conjecture were established by Lagrange. Joseph Louis Lagrange Joseph Louis Lagrange (January 25, 1736 – April 10, 1813; born Giuseppe Luigi Lagrangia in Turin, Lagrange moved to Paris (1787) and became a French citizen, adopting the French translation of his name, Joseph Louis Lagrange) was an Italian-French mathematician and astronomer who made important contributions to...

References

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