# 5

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## Number

$5$ (five) is:

The $3$rd prime number.

The only known odd untouchable number, and probably the only one.

### $1$st Term

The $1$st power of $5$ after the zeroth $1$:
$5 = 5^1$

The $1$st Pythagorean prime, and thus, from Fermat's Two Squares Theorem, the sum of two squares uniquely:
$5 = 4 \times 1 + 1 = 2^2 + 1^2$

The $1$st prime number of the form $6 n - 1$:
$5 = 6 \times 1 - 1$

The $1$st prime number of the form $n! - 1$ for integer $n$:
$5 = 3! - 1$
where $n!$ denotes $n$ factorial

The first prime number of the form $\displaystyle \sum_{k \mathop = 0}^{n - 1} \paren {-1}^k \paren {n - k}!$:
$5 = 3! - 2! + 1!$

The $1$st safe prime:
$5 = 2 \times 2 + 1$

The length of the hypotenuse of the smallest Pythagorean triangle:
$3 - 4 - 5$ triangle

The lower and upper ends of the $1$st record-breaking gap between twin primes, which in this case is no gap at all:
$5 - 5 = 0$

The $1$st of the smallest sequence of both $4$ and $5$ prime numbers in arithmetic sequence:
$5$, $11$, $17$, $23$
$5$, $11$, $17$, $23$, $29$

The $1$st Thabit number after the zeroth: $2$, and $2$nd Thabit prime:
$5 = 3 \times 2^1 - 1$

The $1$st primorial prime:
$5 = p_2 \# - 1 = 3 \# - 1 = 2 \times 3 - 1$

The $1$st positive integer $n$ such that no factorial of an integer can end with $n$ zeroes

The $1$st balanced prime:
$5 = \dfrac {3 + 7} 2$

The $1$st Wilson prime:
$5^2 \divides \paren {5 - 1}! + 1 = 25$

The $1$st of $5$ primes of the form $2 x^2 + 5$:
$2 \times 0^2 + 5 = 5$ (Next)

### $2$nd Term

The $2$nd pentagonal number after $1$:
$5 = 1 + 4 = \dfrac {2 \paren {2 \times 3 - 1} } 2$

The $2$nd pentagonal number after $1$ which is also palindromic:
$5 = 1 + 4 = \dfrac {2 \paren {2 \times 3 - 1} } 2$

The $2$nd element of the $1$st pair of twin primes, with $3$
Also the $1$st element of the $2$nd pair of twin primes, with $7$
Hence the only element of $2$ pairs of twin primes:
$\tuple {3, 5}$ and $\tuple {5, 7}$

The $2$nd square pyramidal number after $1$:
$5 = 1 + 4 = \dfrac {2 \paren {2 + 1} \paren {2 \times 2 + 1} } 6$

The $2$nd pentatope number after $1$:
$5 = 1 + 4 = \dfrac {2 \paren {2 + 1} \paren {2 + 2} \paren {2 + 3} } {24}$

The $2$nd automorphic number after $1$:
$5^2 = 2 \mathbf 5$

The $2$nd Fermat number, and thus Fermat prime, after $3$:
$5 = 2^{\paren {2^1} } + 1$

The $2$nd untouchable number after $2$

The $2$nd term of the $3$-Göbel sequence after $1$, $2$:
$5 = \paren {1 + 1^3 + 2^3} / 2$

The $2$nd $n$ after $4$ such that $n! + 1$ is square: see Brocard's Problem

The $2$nd prime $p$ such that $p \# - 1$, where $p \#$ denotes primorial (product of all primes up to $p$) of $p$, is prime, after $3$:
$5 \# - 1 = 2 \times 3 \times 5 - 1 = 29$

The $2$nd number such that $2 n^2 - 1$ is square, after $1$:
$2 \times 5^2 - 1 = 2 \times 25 - 1 = 49 = 7^2$

The $2$nd prime number after $2$ which divides the sum of all smaller primes:
$1 \times 5 = 5 = 2 + 3$

The $2$nd Sierpiński number of the first kind after $2$:
$5 = 2^2 + 1$

The $2$nd prime Sierpiński number of the first kind after $2$:
$5 = 2^2 + 1$

The $2$nd prime number of the form $n^2 + 1$ after $2$:
$5 = 2^2 + 1$

The $2$nd integer after $2$ at which the prime number race between primes of the form $4 n + 1$ and $4 n - 1$ are tied

The $2$nd of $3$ primes of the form $2 x^2 + 3$:
$2 \times 1^2 + 3 = 5$ (Previous  ... Next)

The $2$nd tri-automorphic number after $2$:
$5^2 \times 3 = 7 \mathbf 5$

The $2$nd after $2$ of $4$ integers whose letters, when spelt in French, are in alphabetical order:
cinq

The $2$nd prime number after $3$ which is palindromic in both decimal and binary:
$5_{10} = 101_2$

The $2$nd of $6$ integers after $2$ which cannot be expressed as the sum of distinct triangular numbers

The $2$nd odd number after $1$ which cannot be expressed as the sum of an integer power and a prime number

### $3$rd Term

The $3$rd (trivial, $1$-digit, after $2$, $3$) palindromic prime

The $3$rd Sophie Germain prime after $2$, $3$:
$2 \times 5 + 1 = 11$, which is prime

The index of the $3$rd Mersenne prime after $2$, $3$:
$M_5 = 2^5 - 1 = 31$

The $3$rd generalized pentagonal number after $1$, $2$:
$5 = \dfrac {2 \paren {3 \times 2 - 1} } 2$

The $3$rd trimorphic number after $1$, $4$:
$5^3 = 12 \mathbf 5$

The $3$rd Catalan number after $(1)$, $1$, $2$:
$5 = \dfrac 1 {3 + 1} \dbinom {2 \times 3} 3 = \dfrac 1 4 \times 20$

The $3$rd term of Göbel's sequence after $1$, $2$, $3$:
$5 = \paren {1 + 1^2 + 2^2 + 3^2} / 3$

The $3$rd Fibonacci prime after $2$, $3$

The $3$rd and last Fibonacci number after $0$, $1$ which equals its index

The $3$rd permutable prime after $2$, $3$

The $3$rd prime $p$ such that $p \# + 1$, where $p \#$ denotes primorial (product of all primes up to $p$) of $p$, is prime, after $2$, $3$:
$5 \# + 1 = 2 \times 3 \times 5 + 1 = 31$

The $3$rd of the lucky numbers of Euler after $2$, $3$:
$n^2 + n + 5$ is prime for $0 \le n < 4$

The $1$st element of the $3$rd pair of consecutive integers whose product is a primorial:
$5 \times 6 = 30 = 5 \#$

The $3$rd Bell number after $(1)$, $1$, $2$

The $3$rd (trivially) two-sided prime after $2$, $3$

The $3$rd prime number after $2$, $3$ consisting (trivially) of a string of consecutive ascending digits

The $3$rd non-negative integer $n$ after $0$, $1$ such that the Fibonacci number $F_n$ ends in $n$

The $3$rd integer $n$ after $3$, $4$ 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:
$5! - 4! + 3! - 2! + 1! = 101$

The $3$rd odd positive integer after $1$, $3$ such that all smaller odd integers greater than $1$ which are coprime to it are prime

The $3$rd odd positive integer after $1$, $3$ that cannot be expressed as the sum of exactly $4$ distinct non-zero square numbers all of which are coprime

### $4$th Term

The $4$th integer after $0$, $1$, $3$ which is palindromic in both decimal and binary:
$5_{10} = 101_2$

The $4$th after $1$, $2$, $4$ of the $24$ positive integers which cannot be expressed as the sum of distinct non-pythagorean primes.

The index of the $4$th Mersenne number after $1$, $2$, $3$ which Marin Mersenne asserted to be prime

The number of distinct free tetrominoes

### $5$th Term

The $5$th Fibonacci number after $1$, $1$, $2$, $3$:
$5 = 2 + 3$

The $5$th of the (trivial $1$-digit) pluperfect digital invariants after $1$, $2$, $3$, $4$:
$5^1 = 5$

The $5$th of the (trivial $1$-digit) Zuckerman numbers after $1$, $2$, $3$, $4$:
$5 = 1 \times 5$

The $5$th of the (trivial $1$-digit) harshad numbers after $1$, $2$, $3$, $4$:
$5 = 1 \times 5$

### $6$th Term

The $6$th integer $n$ after $0$, $1$, $2$, $3$, $4$ such that $2^n$ contains no zero in its decimal representation:
$2^5 = 32$

The $6$th integer $n$ after $0$, $1$, $2$, $3$, $4$ such that $5^n$ contains no zero in its decimal representation:
$5^5 = 3125$

The $6$th integer $n$ after $0$, $1$, $2$, $3$, $4$ such that both $2^n$ and $5^n$ have no zero digits:
$2^5 = 32$, $5^5 = 3125$

The $6$th integer after $0$, $1$, $2$, $3$, $4$ which is (trivially) the sum of the increasing powers of its digits taken in order:
$5^1 = 5$

### Miscellaneous

The magic constant of a magic square of order $2$ (if it were to exist), after $1$:
$5 = \displaystyle \dfrac 1 2 \sum_{k \mathop = 1}^{2^2} k = \dfrac {2 \paren {2^2 + 1} } 2$

## Historical Note

The number $5$ was the number associated by the Pythagoreans with marriage, the sum of $2$, the first female number, and $3$, the first male number.

According to Plutarch, the Pythagoreans also associated $5$ with nature, on account of its automorphic property.

The number $5$ is expressed in Roman numerals as $\mathrm V$.

It has been suggested that this originates from the shape of one hand held up showing the full $5$ fingers.

## Linguistic Note

Words derived from or associated with the number $5$ include:

pentathlon: a competition comprising $5$ athletic events