5

Definition
$5$ (Five) is:


 * The third prime number after $2, 3$


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


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


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


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


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


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


 * Probably the only odd untouchable number.


 * The $5$th Fibonacci number:
 * $1, 1, 2, 3, 5, \ldots$


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


 * 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 $2 x^2 + 3$:
 * $2 \times 1^2 + 3 = 5$

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