Definition:Square Number

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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:

triangular numbers
square numbers
pentagonal numbers
hexagonal numbers

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.