Polygonal Number

A polygonal number is a type of figurate number that is a generalization of triangular, square, etc., to a polygon with sides, where is an integer. The above diagrams graphically illustrate the process by which the polygonal numbers are built up. Starting with the th triangular number , then

(1)

Now note that

(2)

gives the th square number,

(3)

gives the th pentagonal number, and so on. The general polygonal number can be written in the form

			(4)
			(5)

where is the th -gonal number, with a nonnegative integer (Savin 2000). For example, taking in (5) gives a triangular number, gives a square number, etc.

Polygonal numbers are implemented in the Wolfram Language as PolygonalNumber[r, n], while PolygonalNumber[n] gives the th triangular number.

Call a positive integer -highly polygonal if it is -gonal for at least distinct values of in the range from 3 to a specified limit. For , the first few 2-highly polygonal numbers are 1, 6, 9, 10, 12, 15, 16, 21, 28, ... (OEIS A090428). Similarly, the first few 3-highly polygonal numbers for are 1, 15, 36, 45, 325, 561, 1225, 1540, 3025, ... (OEIS A062712). There are no 4-highly polygonal numbers of this type less than except for 1.

The generating function for the -gonal numbers is

(6)

for .

Fermat proposed that every positive integer is expressible as at most -gonal numbers (Fermat's polygonal number theorem). Fermat claimed to have a proof of this result, although this proof has never been found. Jacobi, Lagrange (in 1772), and Euler all proved the square case, and Gauss proved the triangular case in 1796. In 1813, Cauchy proved the proposition in its entirety.

A positive integer can be tested for being -gonal using the identity

(7)

so must be a perfect square. Taking to be its nonnegative square root and solving

(8)

gives

(9)

The number is -gonal iff this value of is a positive integer. Being a perfect square alone is not sufficient. For , an -gonal number is equal to the sum of the -gonal number of the same index and the triangular number of the previous index.