# 4

## Number

$4$ (**four**) is:

- $2^2$

- The only number which equals the number of letters in its name (
**four**) when written in the English language

- The only composite number $n$ such that $n \nmid \paren {n - 1}!$

- The only number $n$ such that the alternating group $A_n$ is not simple.

### $1$st Term

- The $1$st semiprime:
- $4 = 2 \times 2$

- The $1$st power of $4$ after the zeroth $1$:
- $4 = 4^1$

- The $1$st Fermat pseudoprime to base $5$:
- $5^4 \equiv 5 \pmod 4$

- The $1$st $n$ such that $n! + 1$ is square: see Brocard's Problem

- The $1$st positive integer $n$ such that $n - 2^k$ is prime for all $k$.

- The $1$st (with $121$) of the $2$ square numbers which are $4$ less than a cube:
- $4 = 2^2 = 2^3 - 4$

- The $1$st in the sequence formed by adding the squares of the first $n$ primes:
- $4 = \displaystyle \sum_{i \mathop = 1}^1 {p_i}^2 = 2^2$

- The $1$st Smith number:
- $4 = 2 + 2$

- The smallest positive integer which can be expressed as the sum of $2$ distinct lucky numbers in $1$ different way:
- $4 = 1 + 3$

### $2$nd Term

- The $2$nd square number, and the first square of a prime number:
- $4 = 2 \times 2 = 2^2 = 1 + 3$

- The $2$nd square number after $1$ to be the divisor sum value of some (strictly) positive integer:
- $4 = \map {\sigma_1} 3$

- The $2$nd tetrahedral number after $1$:
- $4 = 1 + 3 = \dfrac {2 \paren {2 + 1} \paren {2 + 2} } 6$

- The $2$nd after $1$ of the $3$ tetrahedral numbers which are also square

- The $2$nd trimorphic number after $1$:
- $4^3 = 6 \mathbf 4$

- The $2$nd power of $2$ after $(1)$, $2$:
- $4 = 2^2$

- The $2$nd powerful number after $1$

- The $2$nd integer $m$ after $3$ such that $m! - 1$ (its factorial minus $1$) is prime:
- $4! - 1 = 24 - 1 = 23$

- The $2$nd integer $n$ after $3$ such that $m = \displaystyle \sum_{k \mathop = 0}^{n - 1} \paren {-1}^k \paren {n - k}! = n! - \paren {n - 1}! + \paren {n - 2}! - \paren {n - 3}! + \cdots \pm 1$ is prime:
- $4! - 3! + 2! - 1! = 17$

- The $2$nd even number after $2$ which cannot be expressed as the sum of $2$ composite odd numbers.

- The $2$nd even integer after $2$ that cannot be expressed as the sum of $2$ prime numbers which are each one of a pair of twin primes.

- The $2$nd of $35$ integers less than $91$ to which $91$ itself is a Fermat pseudoprime:
- $3$, $4$, $\ldots$

- The number of different binary operations with an identity element that can be applied to a set with $2$ elements

### $3$rd Term

- The $3$rd Lucas number after $(2)$, $1$, $3$:
- $4 = 1 + 3$

- The $3$rd highly composite number after $1$, $2$:
- $\map {\sigma_0} 4 = 3$

- The $3$rd superabundant number after $1$, $2$:
- $\dfrac {\map {\sigma_1} 4} 4 = \dfrac 7 4 = 1 \cdotp 75$

- The $3$rd almost perfect number after $1$, $2$:
- $\map {\sigma_1} 4 = 7 = 8 - 1$

- The $3$rd after $1$, $2$ of the $24$ positive integers which cannot be expressed as the sum of distinct non-pythagorean primes.

- The $3$rd of the $5$ known powers of $2$ whose digits are also all powers of $2$:
- $1$, $2$, $4$, $\ldots$

- The $3$rd positive integer after $2$, $3$ which cannot be expressed as the sum of distinct pentagonal numbers

### $4$th Term

- The $4$th Ulam number after $1$, $2$, $3$:
- $4 = 1 + 3$

- The $4$th highly abundant number after $1$, $2$, $3$:
- $\map {\sigma_1} 4 = 7$

- The $4$th (strictly) positive integer after $1$, $2$, $3$ which cannot be expressed as the sum of exactly $5$ non-zero squares.

- The $4$th integer after $0$, $1$, $2$ which is palindromic in both decimal and ternary:
- $4_{10} = 11_3$

- The $4$th of the trivial $1$-digit pluperfect digital invariants after $1$, $2$, $3$:
- $4^1 = 4$

- The $4$th positive integer after $1$, $2$, $3$ such that all smaller positive integers coprime to it are prime

- The $4$th of the (trivial $1$-digit) Zuckerman numbers after $1$, $2$, $3$:
- $4 = 1 \times 4$

- The $4$th of the (trivial $1$-digit) harshad numbers after $1$, $2$, $3$:
- $4 = 1 \times 4$

- The $4$th after $1$, $2$, $3$ of $21$ integers which can be represented as the sum of two primes in the maximum number of ways

### $5$th Term

- The $5$th integer $n$ after $0$, $1$, $2$, $3$ such that $2^n$ contains no zero in its decimal representation:
- $2^4 = 16$

- The $5$th integer $n$ after $0$, $1$, $2$, $3$ such that $5^n$ contains no zero in its decimal representation:
- $5^4 = 625$

- The $5$th integer $n$ after $0$, $1$, $2$, $3$ such that both $2^n$ and $5^n$ have no zeroes:
- $2^4 = 16$, $5^4 = 625$

- The $5$th number after $0$, $1$, $2$, $3$ which is (trivially) the sum of the increasing powers of its digits taken in order:
- $4^1 = 4$

### Miscellaneous

- The number of faces and vertices of a tetrahedron

- The number of faces which meet at each vertex of a regular octahedron

- The number of dimensions in Einstein's space-time

- There exist exactly $4$ Kepler-Poinsot polyhedra

- Every positive integer can be expressed as the sum of at most $4$ squares.

## Also see

- Hyperbola can be Drawn through Four Non-Collinear Points
- Plane Figure with Bilateral Symmetry about Two Lines has 4 Congruent Parts
- Lagrange's Four Square Theorem, also represented as Hilbert-Waring Theorem for $2$nd Powers
- Ferrari's Method
- Four Color Theorem
- Squares which are 4 Less than Cubes
- Four Kepler-Poinsot Polyhedra
- Brocard's Problem
- Divisibility of n-1 Factorial by Composite n
- Alternating Group is Simple except on 4 Letters

### Previous in Sequence: $1$

*Previous ... Next*: Trimorphic Number*Previous ... Next*: Powerful Number*Previous ... Next*: Squares with No More than 2 Distinct Digits*Previous ... Next*: Tetrahedral Number*Previous ... Next*: Sequence of Powers of 4*Previous ... Next*: Square Numbers which are Divisor Sum values*Previous ... Next*: Count of Binary Operations with Identity*Previous ... Next*: Powers of 2 which are Sum of Distinct Powers of 3*Previous ... Next*: Square and Tetrahedral Numbers

### Previous in Sequence: $2$

*Previous ... Next*: Superabundant Number*Previous ... Next*: Highly Composite Number*Previous ... Next*: Positive Even Integers not Expressible as Sum of 2 Composite Odd Numbers

*Previous ... Next*: Almost Perfect Number*Previous ... Next*: Palindromes in Base 10 and Base 3*Previous ... Next*: Powers of 2 whose Digits are Powers of 2*Previous ... Next*: Sequence of Powers of 2

### Previous in Sequence: $3$

*Previous ... Next*: Pluperfect Digital Invariant*Previous ... Next*: Zuckerman Number*Previous ... Next*: Harshad Number*Previous ... Next*: Powers of 2 with no Zero in Decimal Representation*Previous ... Next*: Powers of 5 with no Zero in Decimal Representation*Previous ... Next*: Powers of 2 and 5 without Zeroes*Previous ... Next*: Numbers which are Sum of Increasing Powers of Digits*Previous ... Next*: Sum of Sequence of Alternating Positive and Negative Factorials being Prime*Previous ... Next*: Integers whose Number of Representations as Sum of Two Primes is Maximum

*Previous ... Next*: Ulam Number*Previous ... Next*: Highly Abundant Number*Previous ... Next*: Integer not Expressible as Sum of 5 Non-Zero Squares*Previous ... Next*: Sequence of Integers whose Factorial minus 1 is Prime*Previous ... Next*: Integers such that all Coprime and Less are Prime

*Previous ... Next*: Lucas Number*Previous ... Next*: Numbers not Expressible as Sum of Distinct Pentagonal Numbers

### Next in Sequence: $5$

### Next in Sequence: $6$

### Next in Sequence: $7$

### Next in Sequence: $9$

### Next in Sequence: $10$ and above

*Next*: Smallest Sum of 2 Lucky Numbers in n Ways*Next*: Sum of Sequence of Squares of Primes*Next*: Smith Number*Next*: Numbers of Primes with at most n Digits*Next*: Fermat Pseudoprime to Base 5

## Historical Note

The number **$4$**, as was **$8$**, was associated by the Pythagoreans with the concept of **justice**, being evenly balanced: $4 = 2 + 2$, where $2$ is the principle of **diversity**.

Throughout history, the number **$4$** has been regarded with particular significance.

There were originally believed to be $4$ elements out of which everything was formed:

There are $4$ humours:

**Sanguine**,**Melacholic**,**Choleric**and**Phlegmatic**.

There are $4$ cardinal points of the compass:

**North**,**East**,**South**and**West**.

There are $4$ seasons of the year:

**Spring**,**Summer**,**Autumn**and**Winter**.

In the Old Testament, there were $4$ rivers which watered the Garden of Eden:

**Pishon**,**Gihon**, the**Tigris**, and the**Euphrates**.

In Islam, these rivers are:

**Saihan**(**Syr Darya**),**Jaihan**(**Amu Darya**),**Furat**(**Euphrates**) and**Nil**(**Nile**).

In the New Testament, there are $4$ Gospels:

**Matthew**,**Mark**,**Luke**and**John**.

The most aesthetically pleasing musical intervals are those whose frequencies are associated with the ratio of $1 : 4$.

## Linguistic Note

Words derived from or associated with the number $4$ include:

**quadruped**: an animal with $4$ feet

## Sources

- 1986: David Wells:
*Curious and Interesting Numbers*... (previous) ... (next): $4$ - 1997: David Wells:
*Curious and Interesting Numbers*(2nd ed.) ... (previous) ... (next): $4$

- Trimorphic Numbers/Examples
- Powerful Numbers/Examples
- Tetrahedral Numbers/Examples
- Powers of 4/Examples
- Count of Binary Operations with Identity/Examples
- Superabundant Numbers/Examples
- Highly Composite Numbers/Examples
- Almost Perfect Numbers/Examples
- Powers of 2/Examples
- Pluperfect Digital Invariants/Examples
- Zuckerman Numbers/Examples
- Harshad Numbers/Examples
- Ulam Numbers/Examples
- Highly Abundant Numbers/Examples
- Lucas Numbers/Examples
- Semiprimes/Examples
- Square Numbers/Examples
- Smith Numbers/Examples
- Fermat Pseudoprimes/Examples
- Specific Numbers
- 4