**Polygonal Numbers**

Polygonal numbers represent an ancient link between geometry and number theory, and can be traced back to the Greeks (Pythagoras and his followers, in fact).

A good question is, “Can we find a formula for the *n*th polygonal number?”

A good answer is, “Yes.” To do so requires the so-called “method of finite differences”. Well, it doesn’t really “require” it…

**Some Properties of
Polygonal Numbers (that the Greeks knew about)**

Twice any triangular number is an **oblong** number.

The *n*th square
number exceeds its predecessor by the sum of their two roots (Fibonacci).

The *n*th square
number is the sum of the first *n* odd
integers (Pythagoras).

Eight times any triangular number plus one is a square number (Pythagoras).

Any square number is the sum of two consecutive triangular
numbers (Nichomachus, *Introduction to
Arithmetic*).

Any pentagonal number is the sum of a square number and triangular number (Nichomachus).

There exists an “odd number triangle cubic sum mystery”.

Fermat once conjectured that every positive integer is the sum of at most 3 triangular numbers, or at most 4 square numbers, etc…

**Final Thoughts on
Polygonal Numbers**

Find a formula for the *n*th
*m*-gonal number .

Show that if denotes the *n*th triangular number, then the following numbers are also
triangular: . Can you generalize?