# 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:

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 PolygonalNumber 2013-03-22 13:55:38 2013-03-22 13:55:38 mathcam (2727) mathcam (2727) 5 mathcam (2727) Definition msc 11D85 msc 11D09 figurate number pentagonal number