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 attained significance throughout history in part because typical humans have five digits on each hand.
Five is the third smallest prime number.[1] Five is the second Fermat prime,[1] the third Sophie Germain prime,[1] the first safe prime, the third Catalan number,[2] and the third Mersenne prime exponent.[3] Five is the first Wilson prime and the third factorial prime, also an alternating factorial.[4] Five is the first good prime.[5] It is an Eisenstein prime with no imaginary part and real part of the form 3p − 1.[1] It is also the only number that is part of more than one pair of twin primes. Five is also a square pyramidal number, a super-prime, and a congruent number.[6]
Since 5 can be written as 22 + 1, five is classified as a Fermat prime. There are a total of five known Fermat primes, which also include 3, 17, 257, and 65537.[7] The sum of the first 3 Fermat primes (3, 5, and 17) yields 25, or five squared, while 257 is itself the 55th prime. Because 5 is a Fermat prime, a regular polygon with 5 sides (a regular pentagon) is constructible with compass and an unmarked straightedge. Figurate numbers representing pentagons are called pentagonal numbers.
Five is conjectured to be the only odd untouchable number,[8] and if this is the case then five will be the only odd prime number that is not the base of an aliquot tree.
Five is also the only prime that is the sum of two consecutive primes, namely 2 and 3, with these indeed being the only possible set of two consecutive primes.
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. 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.[9]