5

5 (five) is a number, numeral and digit. It is the natural number, and cardinal number, following 4 and preceding 6, and is a prime number. It has garnered attention throughout history in part because distal extremities in humans typically contain five digits.

${\displaystyle 5}$ is the third smallest prime number, and the second super-prime.^[1] It is the first safe prime,^[2] the first good prime,^[3] the first balanced prime,^[4] and the first of three known Wilson primes.^[5] Five is the second Fermat prime^[1] and the third Mersenne prime exponent,^[6] as well as the third Catalan number,^[7] and the third Sophie Germain prime.^[1] Notably, 5 is equal to the sum of the only consecutive primes, 2 + 3, and is the only number that is part of more than one pair of twin primes, (3, 5) and (5, 7).^[8][9] It is also part of the first pair of sexy primes, with the fifth prime number and first prime repunit in decimal, 11.^[10] Five is the third factorial prime,^[11] an alternating factorial,^[12] and an Eisenstein prime with no imaginary part and real part of the form − .^[1] In particular, five is the first congruent number, since it is the length of the hypotenuse of the smallest integer-sided right triangle.^[13]${\displaystyle 3p}$${\displaystyle 1}$

Five is the second Fermat prime of the form + , and more generally the second Sierpiński number of the first kind, + .^[14] There are a total of five known Fermat primes, which also include 3, 17, 257, and 65537.^[15] The sum of the first three Fermat primes, 3, 5 and 17, yields 25 or 5², while 257 is the 55th prime number. Combinations from these five Fermat primes generate 31 polygons with an odd number of sides that can be ${\displaystyle 2^{2^{n}}}$${\displaystyle 1}$${\displaystyle n^{n}}$${\displaystyle 1}$constructed purely with a compass and straight-edge, which includes the five-sided regular pentagon. Apropos, 31 is also equal to the sum of the maximum number of areas inside a circle that are formed from the sides and diagonals of the first five -sided polygons, and equal to the maximum number of areas formed by a six-sided polygon; per Moser's circle problem.^[16]${\displaystyle n}$

The number 5 is the fifth Fibonacci number, being 2 plus 3.^[1] It is the only Fibonacci number that is equal to its position aside from 1, which is both the first and second Fibonacci numbers. Five is also a Pell number and a Markov number, appearing in solutions to the Markov Diophantine equation: (1, 2, 5), (1, 5, 13), (2, 5, 29), (5, 13, 194), (5, 29, 433), ... (OEIS: A030452 lists Markov numbers that appear in solutions where one of the other two terms is 5). Whereas 5 is unique in the Fibonacci sequence, in the Perrin sequence 5 is both the fifth and sixth Perrin numbers.^[17]

The first Pythagorean triple, with a hypotenuse of ${\displaystyle 5}$

Illustration by Leonardo da Vinci of a regular dodecahedron, from Luca Pacioli's Divina proportione

The five sensory organ modalities, with touch represented by the hand's tactility

International maritime signal flag for 5

The fives of all four suits in playing cards