for integers and . A “generalized polygonal number” is any value of for some integer and any . For fixed , is called a -gonal or -polygonal number. For , we speak of a triangular number, a square number or a square, a pentagonal number, and so on.
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 .
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 , any integer 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.
|Date of creation||2013-03-22 13:55:38|
|Last modified on||2013-03-22 13:55:38|
|Last modified by||mathcam (2727)|