The positive integers are the numbers used for counting, and their use as such dates back to the dawn of civilization. No one knows who first became aware of the abstract concept of, say, “seven,” that applies to seven goats, seven trees, seven nights, or any set of seven objects. The counting numbers, along with their negatives and zero, constitute the integers and lie at the heart of mathematics. Thus it is appropriate that we begin with some theorems and proofs about them.

In this chapter we present a variety of results about the integers. Many concern special subsets of the integers, such as squares, triangular numbers, Fibonacci numbers, primes, and perfect numbers. While many of the simpler results can be proven algebraically or by induction, when possible we prefer to present proofs with a visual element. We begin with integers that count objects in sets with a geometric pattern and some identities for them.

Figurate numbers

The idea of representing a number by points in the plane (or perhaps pebbles on the ground) dates back at least to ancient Greece. When the representation takes the shape of a polygon such as a triangle or a square, the number is often called a figurate number. We begin with some theorems and proofs about the simplest figurate numbers: triangular numbers and squares.