Definition:Square Number
This page is about square number. For other uses, see square.
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.
Definition 2
- $S_n = \begin {cases} 0 & : n = 0 \\ S_{n - 1} + 2 n - 1 & : n > 0 \end {cases}$
Definition 3
- $\ds S_n = \sum_{i \mathop = 1}^n \paren {2 i - 1} = 1 + 3 + 5 + \cdots + \paren {2 n - 1}$
Definition 4
- $\forall n \in \N: S_n = \map P {4, n} = \begin{cases} 0 & : n = 0 \\ \map P {4, n - 1} + 2 \paren {n - 1} + 1 & : n > 0 \end{cases}$
where $\map P {k, n}$ 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:
- $0, 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, \ldots$
Also known as
A square number is often referred to as a square.
A square number is also often referred to as a perfect square, but this could cause confusion with the concept of perfect number, so its use is discouraged on $\mathsf{Pr} \infty \mathsf{fWiki}$.
In fact it is prime.mover's opinion that perfect square is so utterly bletheringly pointlessly stupid that he has difficulty wondering whether it's worth carrying on sharing a universe with the utter imbeciles who continue to think it's worthwhile to try and defend its use.
This usage may in fact be regional.
Also see
- Odd Number Theorem which shows that $\ds n^2 = \sum_{j \mathop = 1}^n \paren {2 j - 1}$
- 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
- 1989: Ephraim J. Borowski and Jonathan M. Borwein: Dictionary of Mathematics ... (previous) ... (next): figurate numbers
- 1989: Ephraim J. Borowski and Jonathan M. Borwein: Dictionary of Mathematics ... (previous) ... (next): perfect square or square number
- 1992: George F. Simmons: Calculus Gems ... (previous) ... (next): Chapter $\text {A}.2$: Pythagoras (ca. $\text {580}$ – $\text {500}$ B.C.)
- 1992: George F. Simmons: Calculus Gems ... (previous) ... (next): Chapter $\text {A}.13$: Fermat ($\text {1601}$ – $\text {1665}$)
- 1997: David Wells: Curious and Interesting Numbers (2nd ed.) ... (previous) ... (next): Glossary
- 2014: Christopher Clapham and James Nicholson: The Concise Oxford Dictionary of Mathematics (5th ed.) ... (previous) ... (next): perfect square
- 2021: Richard Earl and James Nicholson: The Concise Oxford Dictionary of Mathematics (6th ed.) ... (previous) ... (next): perfect square