Definition:Square Number

Definition

An integer $n$ is defined as square iff $\exists m \in \Z: n = m^2$.

It is also (in the context of polygonal numbers) called a square number.

For emphasis, such a number is sometimes referred to as a perfect square, but this could cause confusion with the concept of perfect number, so its use is discouraged.

The sequence of square numbers, for $n \in \Z: n \ge 0$ begins:

$0, 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, \ldots$

This sequence is A000290 in the On-Line Encyclopedia of Integer Sequences (N. J. A. Sloane (Ed.), 2008).

Also see the Odd Number Theorem for a well-known recurrence relation defining the square numbers.

Euclid's Definition

As Euclid defined it:

A square number is equal multiplied by equal, or a number which is contained by two equal numbers.

(The Elements: Book VII: Definition $18$)

Sources

• Thomas A. Whitelaw: An Introduction to Abstract Algebra (1978)... (previous)... (next): Exercise $2.13$