polygonal number

A polygonal numberMathworldPlanetmath, or figurate number, is any value of the function


for integers n0 and d3. A “generalized polygonal number” is any value of Pd(n) for some integer d3 and any n. For fixed d, Pd(n) is called a d-gonal or d-polygonal number. For d=3,4,5,, we speak of a triangular numberMathworldPlanetmath, a square number or a square, a pentagonal number, and so on.

An equivalentMathworldPlanetmathPlanetmathPlanetmathPlanetmath definition of Pd, by inductionMathworldPlanetmath on n, is:

Pd(n)=Pd(n-1)+(d-2)(n-1)+1   for all n1
Pd(n-1)=Pd(n)+(d-2)(1-n)-1   for all n<0.

From these equations, we can deduce that all generalized polygonal numbers are nonnegative integers. The first two formulasMathworldPlanetmathPlanetmath show that Pd(n) points can be arranged in a set of n nested d-gons, as in this diagram of P3(5)=15 and P5(5)=35.

Polygonal numbers were studied somewhat by the ancients, as far back as the Pythagoreans, but nowadays their interest is mostly historical, in connection with this famous result:

Theorem: For any d3, any integer n0 is the sum of some d d-gonal numbers.

In other words, any nonnegative integer is a sum of three triangular numbers, four squares, five pentagonal numbers, and so on. Fermat made this remarkable statement in a letter to Mersenne. Regrettably, he never revealed the argument or proof that he had in mind. More than a century passed before Lagrange proved the easiest case: Lagrange’s four-square theorem. The case d=3 was demonstrated by Gauss around 1797, and the general case by Cauchy in 1813.

Title polygonal number
Canonical name PolygonalNumber
Date of creation 2013-03-22 13:55:38
Last modified on 2013-03-22 13:55:38
Owner mathcam (2727)
Last modified by mathcam (2727)
Numerical id 5
Author mathcam (2727)
Entry type Definition
Classification msc 11D85
Classification msc 11D09
Synonym figurate number
Defines pentagonal number