Fermat's set of polygons

The Fermat Polygonalzahlensatz is a mathematical theorem from the theory of numbers . It says that every natural number can be represented as a sum of at most n-corner numbers . A well-known special case is the four-squares theorem , according to which every number can be written as the sum of four square numbers. An example: ${\displaystyle n}$

${\displaystyle 310=17^{2}+4^{2}+2^{2}+1^{2}=289+16+4+1}$

The Fermat set of polygons is named after Pierre de Fermat , from whom the following quote comes:

“I was the first to discover the very beautiful and perfectly general theorem that every number is either a triangular number or the sum of two or three triangular numbers; each number is a square number or the sum of two, three, or four square numbers; either a pentagonal number or the sum of two, three, four or five pentagonal numbers; and so on to infinity, regardless of whether it is a question of hexagonal, heptagonal or any polygonal numbers. I cannot give here the evidence which depends on the many and absurd mysteries of numbers; therefore I intend to dedicate an entire book to this subject and, in this part, to make arithmetically astonishing progress over the previously known limits. "

However, Fermat never published such a book. Joseph Louis Lagrange proved the special case of the four-square theorem in 1770 and Carl Friedrich Gauß in 1796 (unpublished, but he gave evidence for the case of the squares and cubes in his Disquisitiones arithmeticae ) and Legendre (1798) the special case for triangular numbers . However, the proof of the complete theorem did not succeed until Augustin Louis Cauchy in 1815. The proof of Cauchy was considered a sensation at the time and made him famous.

Evidence structure

For the proof of Fermat's polygonal number theorem, the proofs of the triangular number theorem and the four-squares theorem are assumed. For the Cauchy lemma is proven, which says that for with and exist with the following properties: ${\displaystyle n>4}$ ${\displaystyle a,b\in \mathbb {N^{u}} }$ ${\displaystyle b^{2}<4a}$ ${\displaystyle 3a<b^{2}+2b+4}$ ${\displaystyle x,y,z,u}$

${\displaystyle a=x^{2}+y^{2}+z^{2}+u^{2}{\text{ und }}b=x+y+z+u}$

With the help of this theorem, Fermat's polygonal number theorem can be proved by establishing conditions under which the assumptions of Cauchy's lemma apply.

Web links

• Eric W. Weisstein : Fermat's Polygonal Number Theorem . In: MathWorld (English).

Individual evidence

1. ^ Leonard Eugene Dickson : History of the Theory of Numbers. Volume 2: Diophantine Analysis. Dover Publications, Mineola NY 2005, ISBN 0-486-44233-0 , p. 6.
2. ↑ Joseph Louis Lagrange: Démonstration d'un théoreme d'Arithmétique. In: Nouveaux Mémoires de l'Académie Royale des Sciences et Belles-Lettres , 1770. Berlin 1772, pp. 123-133.
3. ↑ On July 10, 1796, Gauß wrote in his diary: "EYPHKA num = Δ + Δ + Δ". A proof can be found in Hermann Maser (Ed.): Carl Friedrich Gauss' Investigations on Higher Arithmetic. Berlin: Springer, 1889, pp. 333–334, Art. 293.
4. ^ Augustin Louis Cauchy: Démonstration du théorème général de Fermat sur les nombres polygones. In: Mémoires de la class des sciences mathématiques et physiques de l'Institut de France 14 (1813–1815), pp. 177–220.
5. ↑ Bruno Belhoste: Augustin-Louis Cauchy. A biography. New York: Springer, 1991, p. 46.
6. ^ Melvyn B. Nathanson: A Short Proof of Cauchy's Polygonal Number Theorem . In: Proceedings of the American Mathematical Society . tape 99 , no. 1 , 1987, pp. 22-24 , doi : 10.2307 / 2046263 .