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 squarenumber 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 polygonalnumbers are nonnegative integers.The first two formulas show that $P_{d}(n)$ points can be arranged in aset 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 farback as the Pythagoreans, but nowadays their interestis mostly historical, in connection with this famous result:

Theorem: For any $d\\geq 3$ , any integer $n\\geq 0$ is thesum of some $d$ $d$ -gonal numbers.

In other words, any nonnegative integer is a sum of threetriangular 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 hehad in mind. More than a century passed before Lagrange provedthe easiest case: Lagrange’s four-square theorem. Thecase $d=3$ was demonstrated by Gauss around 1797, and thegeneral case by Cauchy in 1813.