# Category:Square Numbers

This category contains results about Square Numbers.
Definitions specific to this category can be found in Definitions/Square Numbers.

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.

### 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.