# Definition:Square Number

## Contents

## Definition

**Square numbers** are those denumerating a collection of objects which can be arranged in the form of a square.

They can be denoted:

- $S_1, S_2, S_3, \ldots$

### Definition 1

An integer $n$ is classified as a **square number** if and only if:

- $\exists m \in \Z: n = m^2$

where $m^2$ denotes the integer square function.

#### Euclid's Definition

In the words of Euclid:

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

(*The Elements*: Book $\text{VII}$: Definition $18$)

### Definition 2

- $S_n = \begin{cases} 0 & : n = 0 \\ S_{n-1} + 2 n - 1 & : n > 0 \end{cases}$

### Definition 3

- $\displaystyle S_n = \sum_{i \mathop = 1}^n \left({2 i - 1}\right) = 1 + 3 + 5 + \cdots + \left({2 n - 1}\right)$

### Definition 4

- $\forall n \in \N: S_n = P \left({4, n}\right) = \begin{cases} 0 & : n = 0 \\ P \left({4, n - 1}\right) + 2 \left({n - 1}\right) + 1 & : n > 0 \end{cases}$

where $P \left({k, n}\right)$ denotes the $k$-gonal numbers.

## Examples of Square Numbers

The first few square numbers are as follows:

### Sequence of Square Numbers

The sequence of square numbers, for $n \in \Z_{\ge 0}$, 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 known as

A **square number** is often referred to as a **square**.

For emphasis, a **square 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.

## Also see

- Odd Number Theorem which shows that $\displaystyle n^2 = \sum_{j \mathop = 1}^n \paren {2 j - 1}$

- Results about
**square numbers**can be found here.

## Historical Note

**Figurate numbers**, that is:

and so on, were classified and investigated by the Pythagorean school in the $6$th century BCE. This was possibly the first time this had ever been done.

## Sources

- 1978: Thomas A. Whitelaw:
*An Introduction to Abstract Algebra*... (previous) ... (next): Chapter $2$: Some Properties of $\Z$: Exercise $2.13$ - 1986: David Wells:
*Curious and Interesting Numbers*... (previous) ... (next): Glossary - 1992: George F. Simmons:
*Calculus Gems*... (previous) ... (next): Chapter $\text {A}.2$: Pythagoras (ca. $580$ – $500$ B.C.) - 1992: George F. Simmons:
*Calculus Gems*... (previous) ... (next): Chapter $\text {A}.13$: Fermat ($1601$ – $1665$) - 1997: David Wells:
*Curious and Interesting Numbers*(2nd ed.) ... (previous) ... (next): Glossary