PlanetMath: polygonal number

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polygonal number

(Definition)

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

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

for integers $n\ge 0$ and $d\ge 3$ . A “generalized polygonal number” is any value of

for some integer $d\ge 3$ and any $n\in\mathbb{Z}$ . For fixed

is called a

-gonal or

-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 , by induction on , is:

$\displaystyle P_d(0)=0$

$\displaystyle P_d(n)=P_d(n-1)+(d-2)(n-1)+1$ for all $\displaystyle n\ge 1$

$\displaystyle P_d(n-1)=P_d(n)+(d-2)(1-n)-1$ for all $\displaystyle n<0\;.$

From these equations, we can deduce that all generalized polygonal numbers are nonnegative integers. The first two formulas show that

points can be arranged in a set of

nested

-gons, as in this diagram of

and

$\includegraphics{dgon}$

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 $d\ge 3$ , any integer $n\ge 0$ is the sum of some -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 was demonstrated by Gauss around 1797, and the general case by Cauchy in 1813.

"polygonal number" is owned by mathcam. [ owner history (1) ]

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Other names:

figurate number

Also defines:

pentagonal number

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Cross-references: Gauss, Lagrange's four-square theorem, proof, argument, sum, points, equations, induction, equivalent, square, triangular number, number, integers, function
There are 9 references to this entry.

This is version 2 of polygonal number, born on 2003-09-02, modified 2003-09-03.
Object id is 4685, canonical name is PolygonalNumber.
Accessed 9169 times total.

Classification:

AMS MSC:	11D85 (Number theory :: Diophantine equations :: Representation problems)
	11D09 (Number theory :: Diophantine equations :: Quadratic and bilinear equations)

Pending Errata and Addenda

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