Fermat polygonal number theorem
- Fermat polygonal number theorem
In mathematics, the Fermat polygonal number theorem states: every positive integer is a sum of at most -polygonal numbers. That is, every number can be written as the sum of at most three triangular numbers, or four square numbers, or five pentagonal numbers and so on.
An example of the triangular number case would be 17 = 10 + 6 + 1.
A well-known special case of this is Lagrange's four-square theorem, which states that every positive number can be represented as a sum of four squares, for example, 7 = 4 + 1 + 1 + 1.
Joseph Louis Lagrange proved the square case in 1770 and Gauss proved the triangular case in 1796, but the theorem was not resolved until it was finally proven by Cauchy in 1813. Nathanson's proof (see the references) is based on the following lemma due to Cauchy:
For odd positive integers and such that and
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