polygonal number

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

$P_{d}(n)=\frac{(d-2)n^{2}+(4-d)n}{2}$

for integers $n\geq 0$ and $d\geq 3$ . A “generalized polygonal number” is any value of $P_{d}(n)$ for some integer $d\geq 3$ and any $n\in\mathbb{Z}$ . For fixed $d$ , $P_{d}(n)$ is called a $d$ -gonal or $d$ -polygonal number. For $d=3,4,5,\ldots$ , we speak of a triangular number, a square number or a square, a pentagonal number, and so on.

An equivalent definition of $P_{d}$ , by induction on $n$ , is:

$P_{d}(0)=0$

$P_{d}(n)=P_{d}(n-1)+(d-2)(n-1)+1\qquad\text{ for all }n\geq 1$

$P_{d}(n-1)=P_{d}(n)+(d-2)(1-n)-1\qquad\text{ for all }n<0\;.$

From these equations, we can deduce that all generalized polygonal numbers are nonnegative integers. The first two formulas show that $P_{d}(n)$ points can be arranged in a set of $n$ nested $d$ -gons, as in this diagram of $P_{3}(5)=15$ and $P_{5}(5)=35$ .

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 $d\geq 3$ , any integer $n\geq 0$ 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