Category:Definitions/Square Numbers
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This category contains definitions related to Square Numbers.
Related results can be found in Category: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.
(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
- $\ds S_n = \sum_{i \mathop = 1}^n \paren {2 i - 1} = 1 + 3 + 5 + \cdots + \paren {2 n - 1}$
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.
Pages in category "Definitions/Square Numbers"
The following 7 pages are in this category, out of 7 total.