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


 * 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 $m = \ds \sum_{k \mathop = 0}^{n - 1} \paren {-1}^k \paren {n - k}! = n! - \paren {n - 1}! + \paren {n - 2}! - \paren {n - 3}! + \cdots \pm 1$ 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

$4$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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