5

== 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 $\ds \sum_{k \mathop = 0}^{n - 1} \paren {-1}^k \paren {n - k}!$:
::$5 = 3! - 2! + 1!$
: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 $2 x^2 + 5$:
::$2 \times 0^2 + 5 = 5$ ()
=== $2$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 larger element of the $1$st pair of twin primes, with $3$
:Also the smaller 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 Proth prime after $3$:
::$5 = 1 \times 2^2 + 1$
: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 $2 x^2 + 3$:
::$2 \times 1^2 + 3 = 5$ ()
: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 after $1$, $2$ of $6$ integers $n$ such that the alternating group $A_n$ is ambivalent
: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
::$F_5 = 5$
: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) left-truncatable prime after $2$, $3$
:The $3$rd (trivially) right-truncatable prime after $2$, $3$
: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 $m = \ds \sum_{k \mathop = 0}^{n - 1} \paren {-1}^k \paren {n - k}! = n! - \paren {n - 1}! + \paren {n - 2}! - \paren {n - 3}! + \cdots \pm 1$ 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
:The $3$rd minimal prime base $10$ after $2$, $3$
:The $3$rd integer $n$ after $3$, $4$ for which the Ramanujan-Nagell equation $x^2 + 7 = 2^n$ has an integral solution:
::$5^2 + 7 = 32 = 2^5$
:The $3$rd positive integer solution after $1$, $3$ to the Ramanujan-Nagell equation $x^2 + 7 = 2^n$ for integral $n$:
::$5^2 + 7 = 32 = 2^5$
=== $4$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 asserted to be prime
:The number of distinct free tetrominoes
:The number of integer partitions for $4$:
::$\map p 4 = 5$
=== $5$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 $1$, $2$, $3$, $4$:
::$5 = 1 \times 5$
:The $5$th after $1$, $2$, $3$, $4$ of $21$ integers which can be represented as the sum of two primes in the maximum number of ways
=== $6$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 $1$:
::$5 = \ds \dfrac 1 2 \sum_{k \mathop = 1}^{2^2} k = \dfrac {2 \paren {2^2 + 1} } 2$
== Also see ==
* Fibonacci Numbers which equal their Index
* Odd Untouchable Numbers
* Volume of Unit Hypersphere
* Lamé's Theorem
* Abel-Ruffini Theorem
* Number is Sum of Five Cubes
* Conic Section through Five Points
* Five Platonic Solids
* Brocard's Problem
* Five Color Theorem
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=== Previous in sequence: $4$ ===
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=== Next in sequence: $6$ and above ===
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== Historical Note ==
== Linguistic Note ==
== Sources ==
* : $5$
* : $5$
Category:Specific Numbers
Category:5