PolygonalNumber 2003-09-02 17:07:58 2003-09-03 02:12:58 Definition pentagonal number \usepackage{amssymb} \usepackage{amsmath} \usepackage{amsfonts} \usepackage{graphicx} \PMlinkescapeword{connection} \PMlinkescapeword{fixed} \PMlinkescapeword{formulas} 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 $n\ge 0$ and $d\ge 3$. A ``generalized polygonal number'' is any value of $P_d(n)$ for some integer $d\ge 3$ 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: $$P_d(0)=0$$ $$P_d(n)=P_d(n-1)+(d-2)(n-1)+1\qquad\text{ for all }n\ge 1$$ $$P_d(n-1)=P_d(n)+(d-2)(1-n)-1\qquad\text{ for all }n<0\;.$$ 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 $P_5(5)=35$. \begin{center}\includegraphics{dgon}\end{center} 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: \textbf{Theorem: }For any $d\ge 3$, any integer $n\ge 0$ 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.