# 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 $n\ge 0$ 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 $d=3,4,5,\mathrm{\dots}$, 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:

$${P}_{d}(0)=0$$ |

$${P}_{d}(n)={P}_{d}(n-1)+(d-2)(n-1)+1\mathit{\hspace{1em}\hspace{1em}}\text{for all}n\ge 1$$ |

$$ |

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 $d\ge 3$, 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.

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 |