Two squares set

The two-square theorem of Fermat is a mathematical theorem of number theory , he states:

An odd prime number can be written as ${\displaystyle p}$

${\displaystyle p=x^{2}+y^{2}}$

can be expressed with integers and if ${\displaystyle x}$ ${\displaystyle y}$

${\displaystyle p\equiv 1{\pmod {4}}.}$

Prime numbers to which this applies are also called Pythagorean prime numbers .

For example, the prime numbers 5, 13, 17, 29, 37 and 41 are congruent to 1 modulo 4 and they can be written as the sum of two squares as follows:

${\displaystyle 5=1^{2}+2^{2},\quad 13=2^{2}+3^{2},\quad 17=1^{2}+4^{2},\quad 29=2^{2}+5^{2},\quad 37=1^{2}+6^{2},\quad 41=4^{2}+5^{2}.}$

On the other hand, the prime numbers 3, 7, 11, 19, 23 and 31 are congruent to 3 modulo 4 and none can be written as the sum of two squares. This is the easier part of the theorem, it follows immediately from the observation that a square modulo 4 can only be congruent to 0 or 1.

Historical remarks

First is Albert Girard made this observation, he described even all positive integers, not only prime numbers, the two as the sum of squares can be expressed, it was published 1625th The statement that every prime number of the form is the sum of two squares is sometimes called Girard's theorem . This part of the statement as well as the determination of the number of different possibilities to write a given prime power as the sum of two squares was elaborated by Fermat in a letter to Marin Mersenne , dated December 25, 1640. Therefore, this version of the theorem is sometimes also Fermat's Called the Christmas theorem. ${\displaystyle 4n+1}$

Proofs of the two-squares theorem

Usually Fermat did not publish any evidence of his claims, nor did he provide any evidence for the two-squares theorem. A first proof was a lot of effort by the method of infinite descent of Euler found. He first announced it to Goldbach in two letters dated May 6, 1747 and April 12, 1749 , and the full evidence was then published in two articles between 1752 and 1755. Lagrange provided evidence in 1775 through his studies of square shapes . This was simplified by Gauss in his Disquisitiones Arithmeticae . Dedekind provided at least two proofs based on the arithmetic of Gaussian numbers . There is also an elegant proof using the Minkovskian lattice point theorem . Zagier has found a very brief proof, a simplification of an earlier brief proof by Heath-Brown , which in turn was inspired by ideas derived from Lagrange. Only recently D. Christopher found a combinatorial number theoretic proof.

Related results

Fourteen years later, Fermat had announced two related results. In a letter to Blaise Pascal dated September 25, 1654 , he claimed about an odd prime number : ${\displaystyle p}$

• ${\displaystyle p}$is of the form if and only if ${\displaystyle x^{2}+2y^{2}}$ ${\displaystyle p\equiv 1{\mbox{ oder }}p\equiv 3{\pmod {8}}}$
• ${\displaystyle p}$is of the form if and only if . ${\displaystyle x^{2}+3y^{2}}$ ${\displaystyle p\equiv 1{\pmod {3}}}$

He further wrote:

If two prime numbers end in the digits 3 or 7 and are both larger by 3 than a multiple of 4, their product is a sum of a square and five times a square.

In other words, if and are prime numbers of the form or , then is of the form . Euler later expanded this to suggest that ${\displaystyle p}$ ${\displaystyle q}$ ${\displaystyle 20k+3}$ ${\displaystyle 20k+7}$ ${\displaystyle pq}$ ${\displaystyle x^{2}+5y^{2}}$

• ${\displaystyle p}$is of the form if and only if , ${\displaystyle x^{2}+5y^{2}}$ ${\displaystyle p\equiv 1{\mbox{ oder }}p\equiv 9{\pmod {20}}}$
• ${\displaystyle 2p}$is of the form if and only if . ${\displaystyle x^{2}+5y^{2}}$ ${\displaystyle p\equiv 3{\mbox{ oder }}p\equiv 7{\pmod {20}}}$

Both of Fermat's claims, as well as Euler's conjectures, were ultimately proven by Lagrange.

According to the Brahmagupta – Fibonacci identity , the product of two whole numbers, which can both be represented as the sum of two squares, is again a sum of two squares. If one now applies Fermat's two-squares theorem to the prime factorization of a positive number , one recognizes that the sum of two squares can be represented if every prime factor that is congruent to 3 modulo 4 occurs with an even exponent . The reverse also applies here. ${\displaystyle n}$ ${\displaystyle n}$

See also

• Legendre's three-squares theorem
• Lagrangian four-squares theorem
• Sum of squares function
• Lemma of Thue

Individual evidence

1. ↑ Simon Stevin : l'Arithmétique de Simon Stevin de Bruges , annotated by Albert Girard, Leyden 1625, page 622.
2. ^ LE Dickson: History of the Theory of Numbers , Volume II, Chap. VI, p. 227.
3. ^ LE Dickson: History of the Theory of Numbers , Volume II, Chap. VI, p. 228.
4. ↑ De numerus qui sunt aggregata quorum quadratorum. (Novi commentarii academiae scientiarum Petropolitanae 4 (1752/3), 1758, pages 3-40)
5. ↑ Demonstratio theorematis FERMATIANI omnem numerum primum formae 4n + 1 esse summam duorum quadratorum. (Novi commentarii academiae scientiarum Petropolitanae 5 (1754/5), 1760, pages 3-13)
6. ^ D. Zagier: A one-sentence proof that every prime p  ≡ 1 (mod 4) is a sum of two squares , American Mathematical Monthly , 1990, Volume 97, Page 144
7. ^ A. David Christopher. A partition-theoretic proof of Fermat's Two Squares Theorem , Discrete Mathematics (2016), Volume 339, pages 1410-1411.
8. ↑ GH Hardy , EM Wright : An introduction to the theory of numbers , Chapter 20.1, Theorems 367 and 368, Oxford 1938.

literature

• LE Dickson : History of the Theory of Numbers Volume 2. Chelsea Publishing Co., New York 1920
• J. Stillwell : Introduction to Theory of Algebraic Integers by Richard Dedekind , Cambridge University Library, Cambridge University Press 1996, ISBN 0-521-56518-9
• DA Cox : Primes of the Form x ²  + ny ² , Wiley-Interscience 1989, ISBN 0-471-50654-0