Editing Fermat polygonal number theorem

{{short description|Every positive integer is a sum of at most n n-gonal numbers}}
{{distinguish|Fermat's Last Theorem}}

In [[additive number theory]], the '''Fermat polygonal number theorem''' states that every positive integer is a sum of at most {{mvar|n}} [[Polygonal number|{{mvar|n}}-gonal number]]s. That is, every positive integer can be written as the sum of three or fewer [[triangular number]]s, and as the sum of four or fewer [[square number]]s, and as the sum of five or fewer [[pentagonal number]]s, and so on. That is, the {{mvar|n}}-gonal numbers form an [[additive basis]] of order {{mvar|n}}.

==Examples==
Three such representations of the number 17, for example, are shown below:
*17 = 10 + 6 + 1 (''triangular numbers'')
*17 = 16 + 1 (''square numbers'')
*17 = 12 + 5 (''pentagonal numbers'').

==History==
[[File:Eureka Gauss.png|thumb|[[Gauss's diary]] entry related to sum of triangular numbers (1796)]]
The theorem is named after [[Pierre de Fermat]], who stated it, in 1638, without proof, promising to write it in a separate work that never appeared.<ref name="heath">{{harvtxt|Heath|1910}}.</ref>
[[Joseph Louis Lagrange]] proved the [[Lagrange's four-square theorem|square case]] in 1770, which states that every positive number can be represented as a sum of four squares, for example, {{nowrap|1=7 = 4 + 1 + 1 + 1}}.<ref name="heath"/> [[Carl Friedrich Gauss|Gauss]] proved the triangular case in 1796,  commemorating the occasion by writing in [[Gauss's diary|his diary]] the line "[[Eureka (word)|ΕΥΡΗΚΑ!]] {{nowrap|1=num = Δ + Δ + Δ}}",<ref>{{citation|last=Bell|first=Eric Temple|authorlink=Eric Temple Bell|contribution=Gauss, the Prince of Mathematicians|editor-last=Newman|editor-first=James R.|title=The World of Mathematics|volume=I|pages=295–339|publisher=[[Simon & Schuster]]|year=1956}}. Dover reprint, 2000, {{ISBN|0-486-41150-8}}.</ref> and published a proof in his book [[Disquisitiones Arithmeticae]]. For this reason, Gauss's result is sometimes known as the '''Eureka theorem'''.<ref>{{citation
 | last1 = Ono | first1 = Ken
 | last2 = Robins | first2 = Sinai
 | last3 = Wahl | first3 = Patrick T.
 | doi = 10.1007/BF01831114
 | mr = 1336863
 | issue = 1–2
 | journal = [[Aequationes Mathematicae]]
 | pages = 73–94
 | title = On the representation of integers as sums of triangular numbers
 | volume = 50
 | year = 1995| s2cid = 122203472
 }}.</ref> The full polygonal number theorem was not resolved until it was finally proven by [[Cauchy]] in 1813.<ref name="heath"/> The proof of {{harvtxt|Nathanson|1987}} is based on the following lemma due to Cauchy:

For odd positive integers {{mvar|a}} and {{mvar|b}} such that {{math|''b''<sup>2</sup> < 4''a''}} and {{math|3''a'' < ''b''<sup>2</sup> + 2''b'' + 4}} we can find nonnegative integers {{mvar|s}}, {{mvar|t}}, {{mvar|u}}, and {{mvar|v}} such that
{{math|1=''a'' = ''s''<sup>2</sup> + ''t''<sup>2</sup> + ''u''<sup>2</sup> + ''v''<sup>2</sup>}} and {{math|1=''b'' = ''s'' + ''t'' + ''u'' + ''v''}}.

==See also==
* [[Pollock's conjectures]]
* [[Waring's problem]]

==Notes==
{{reflist}}

==References==
* {{mathworld|urlname=FermatsPolygonalNumberTheorem|title=Fermat's Polygonal Number Theorem}}
*{{citation | last = Heath | first = Sir Thomas Little | authorlink = Thomas Little Heath | title = Diophantus of Alexandria; a study in the history of Greek algebra | year = 1910 | publisher = Cambridge University Press | url = https://archive.org/details/diophantusofalex00heatiala | page = 188}}.
*{{citation
 | last = Nathanson | first = Melvyn B. | authorlink = Melvyn B. Nathanson
 | doi = 10.2307/2046263
 | mr = 866422
 | issue = 1
 | journal = Proceedings of the American Mathematical Society
 | pages = 22–24
 | title = A short proof of Cauchy's polygonal number theorem
 | volume = 99
 | year = 1987| jstor = 2046263 }}.
*{{citation
  | last1 =  Nathanson | first1 = Melvyn B. 
  | title = Additive Number Theory The Classical Bases 
  | publisher = [[Springer Science+Business Media|Springer]]
  | location = Berlin
  | year = 1996
  | isbn = 978-0-387-94656-6}}. Has proofs of Lagrange's theorem and the polygonal number theorem.
{{Pierre de Fermat}}

[[Category:Additive number theory]]
[[Category:Analytic number theory]]
[[Category:Figurate numbers]]
[[Category:Theorems in number theory]]