5

Number
$5$ (five) is:


 * The $3$rd prime number after $2, 3$


 * The $2$nd of the $1$st pair of twin primes, with $3$
 * Also the $1$st of the $2$nd pair of twin primes, with $7$
 * Hence the only prime number to be one of $2$ pairs of twin primes.


 * 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 $2$nd pentagonal number after $1$:
 * $5 = 1 + 4 = \dfrac {2 \left({2 \times 3 - 1}\right)} 2$


 * The $2$nd square pyramidal number after $1$:
 * $5 = 1 + 4 = \dfrac {2 \left({2 + 1}\right) \left({2 \times 2 + 1}\right)} 6$


 * The $2$nd pentatope number after $1$:
 * $5 = 1 + 4 = \dfrac {2 \left({2 + 1}\right) \left({2 + 2}\right) \left({2 + 3}\right)} {24}$


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


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


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


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


 * The $2$nd Fermat number, and thus Fermat prime, after $3$:
 * $5 = 2^{\left({2^1}\right)} + 1$


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


 * The only member of $2$ pairs of twin primes:
 * $\left({3, 5}\right)$ and $\left({5, 7}\right)$


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


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


 * The $2$nd untouchable number after $2$.


 * The $3$rd term of Göbel's sequence after $1, 2, 3$:
 * $5 = \left({1 + 1^2 + 2^2 + 3^2}\right) / 3$


 * The $2$nd term of the $3$-Göbel sequence after $1, 2$:
 * $5 = \left({1 + 1^3 + 2^3}\right) / 2$


 * Probably the only odd untouchable number.


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


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


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


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


 * The $2$nd of the sequence of $n$ such that $p_n \# - 1$, where $p_n \#$ denotes primorial of $n$, is prime, after $3$:
 * $p_5 \# - 1 = 2 \times 3 \times 5 - 1 = 29$


 * The $3$rd of the sequence of $n$ such that $p_n \# + 1$, where $p_n \#$ denotes primorial of $n$, is prime, after $2, 3$:
 * $p_5 \# + 1 = 2 \times 3 \times 5 + 1 = 31$


 * 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 first of $5$ primes of the form $2 x^2 + 5$:
 * $2 \times 0^2 + 5 = 5$


 * The $2$nd of $3$ primes of the form $2 x^2 + 3$:
 * $2 \times 1^2 + 3 = 5$


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


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


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


 * 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 $2$nd positive integer after $2$ which divides the sum of all smaller primes:
 * $1 \times 5 = 5 = 2 + 3$


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


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


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


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

Also see

 * Odd Untouchable Numbers
 * Volume of Unit Hypersphere
 * Maximum Number of Steps taken by Euclidean Algorithm
 * Abel-Ruffini Theorem
 * Number is Sum of Five Cubes
 * Conic Section through Five Points
 * Five Platonic Solids