# Category:Square Numbers

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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.

(*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.

## Subcategories

This category has the following 27 subcategories, out of 27 total.

### 1

- 1+2+...+n+(n-1)+...+1 = n^2 (6 P)

### F

- Fermat Number is not Square (3 P)

### I

### L

- Locker Problem (3 P)

### N

### O

- Odd Number Theorem (3 P)

### P

### S

- Square Modulo 3 (4 P)
- Square Numbers/Examples (175 P)
- Sum of Sequence of Squares (10 P)

### T

## Pages in category "Square Numbers"

The following 128 pages are in this category, out of 128 total.

### C

### D

### F

### H

### I

- If Ratio of Square to Number is as between Two Squares then Number is Square
- Index of Square Triangular Number from Preceding
- Integer as Difference between Two Squares
- Integer as Difference between Two Squares/Formulation 1
- Integer as Difference between Two Squares/Formulation 2
- Integer both Square and Triangular
- Integers whose Ratio between Divisor Sum and Phi is Square
- Integers whose Squares end in 444

### N

- Number divides Number iff Square divides Square
- Number does not divide Number iff Square does not divide Square
- Number which is Square and Cube Modulo 7
- Number whose Square and Cube use all Digits Once
- Number whose Square is in 2 Identical Halves
- Numbers not Sum of Square and Prime
- Numbers whose Cube equals Sum of Sequence of that many Squares
- Numbers whose Divisor Sum is Square
- Numbers whose Square is Palindromic with Even Number of Digits
- Numbers whose Squares are Consecutive Odd or Even Integers Juxtaposed
- Numbers whose Squares have Digits which form Consecutive Decreasing Integers
- Numbers whose Squares have Digits which form Consecutive Increasing Integers
- Numbers whose Squares have Digits which form Consecutive Integers

### P

- Palindromic Squares with Even Number of Digits with Non-Palindromic Roots
- Palindromic Squares with Non-Palindromic Roots
- Pandigital Pairs whose Squares are Pandigital
- Parity of Integer equals Parity of its Square
- Parity of Integer equals Parity of its Square/Even
- Parity of Integer equals Parity of its Square/Odd
- Penholodigital Square Equation
- Product of 4 Consecutive Integers is One Less than Square
- Product of Two Triangular Numbers to make Square
- Properties of Family of 333,667 and Related Numbers/Squares

### R

### S

- Sequence of Square Centered Hexagonal Numbers
- Sequence of Square Lucky Numbers
- Sequence of Squares Beginning and Ending with n 4s
- Set of 7 Anagrams which are Square
- Smallest n needing 6 Numbers less than n so that Product of Factorials is Square
- Smallest Non-Palindromic Number with Palindromic Square
- Smallest Pandigital Square
- Smallest Penholodigital Square
- Smallest Square which is Sum of 3 Fourth Powers
- Square and Tetrahedral Numbers
- Square Cullen Numbers
- Square Fibonacci Number
- Square Formed from Sum of 4 Consecutive Binomial Coefficients
- Square Modulo 3
- Square Modulo 4
- Square Modulo 5
- Square Modulo 5/Corollary
- Square Modulo 8
- Square Number ending in 9 Digits in Reverse Order
- Square Number/Examples
- Square Number/Sequence
- Square Numbers which are Divisor Sum values
- Square Numbers which are Sum of Consecutive Powers
- Square Numbers which are Sum of Sequence of Odd Cubes
- Square Numbers whose Divisor Sum is Square
- Square of 1 Less than Number Base
- Square of Odd Multiple of 3 is Difference between Triangular Numbers
- Square of Odd Number as Difference between Triangular Numbers
- Square of Repdigit Number consisting of Instances of 3
- Square of Repdigit Number consisting of Instances of 6
- Square of Repunit times Sum of Digits
- Square of Reversal of Small-Digit Number
- Square of Small Repunit is Palindromic
- Square of Small-Digit Palindromic Number is Palindromic
- Square Product of Three Consecutive Triangular Numbers
- Square Pyramidal Number also Square
- Square Sum of Three Consecutive Triangular Numbers
- Square which is 2 Less than Cube
- Square which is Difference between Square and Square of Reversal
- Square whose Divisor Sum is Cubic
- Squares Ending in Repeated Digits
- Squares equal to Sum of 2 Cubes
- Squares of form 2 n^2 - 1
- Squares which are 4 Less than Cubes
- Squares which are Difference between Two Cubes
- Squares whose Digits can be Separated into 2 other Squares
- Squares whose Digits form Consecutive Decreasing Integers
- Squares whose Digits form Consecutive Increasing Integers
- Squares whose Digits form Consecutive Integers
- Squares with All Odd Digits
- Squares with No More than 2 Distinct Digits
- Sufficient Condition for Square of Product to be Triangular
- Sum of 2 Squares in 2 Distinct Ways
- Sum of 2 Squares in 3 Distinct Ways
- Sum of 3 Squares in 2 Distinct Ways
- Sum of 4 Consecutive Binomial Coefficients forming Square
- Sum of Consecutive Triangular Numbers is Square
- Sum of Sequence of Even Squares
- Sum of Sequence of Squares
- Sum of Sequence of Squares of Primes
- Sum of Squares of Divisors of 24 and 26 are Equal
- Sums of Consecutive Sequences of Squares that equal Squares
- Sums of Partial Sequences of Squares
- Sums of Sequences of Consecutive Squares which are Square