MathDB

Prove that for all

n \ge 3

there are an infinite number of

n

-sided polygonal numbers which are also the sum of two other (not necessarily different)

n

-sided polygonal numbers!
The first

n

-sided polygonal number is

1

. The kth n-sided polygonal number for

k \ge 2

is the number of different points in a figure that consists of all of the regular

n

-sided polygons which have one common vertex, are oriented in the same direction from that vertex and their sides are

\ell

cm long where

1 \le \ell \le k - 1

cm and

\ell

is an integer.
In this figure, what we call points are the vertices of the polygons and the points that break up the sides of the polygons into exactly

1

cm long segments. For example, the first four pentagonal numbers are 1,5,12, and 22, like it is shown in the figure. https://cdn.artofproblemsolving.com/attachments/1/4/290745d4be1888813678127e6d63b331adaa3d.png

Problem Statement