2005 In Music

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.

In mathematics

$5$ is the third smallest prime number, and the second super-prime. It is the first safe prime, the first good prime, the first balanced prime, and the first of three known Wilson primes. Five is the second Fermat prime and the third Mersenne prime exponent, as well as the third Catalan number, and the third Sophie Germain prime. 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). It is also a sexy prime with the fifth prime number and first prime repunit, 11. Five is the third factorial prime, an alternating factorial, and an Eisenstein prime with no imaginary part and real part of the form $3p$ − $1$. In particular, five is the first congruent number, since it is the length of the hypotenuse of the smallest integer-sided right triangle.

Five is the second Fermat prime of the form $2^$+ $1$, and more generally the second SierpiÅ„ski number of the first kind, $n^n$+ $1$. There are a total of five known Fermat primes, which also include 3, 17, 257, and 65537. The sum of the first 3 Fermat primes, 3, 5 and 17, yields 25 or 5², while 257 is the 55th prime number. Combinations from these 5 Fermat primes generate 31 polygons with an odd number of sides that can be construncted 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 $n$-sided polygons, and equal to the maximum number of areas formed by a six-sided polygon; per Moser's circle problem.

The number 5 is the fifth Fibonacci number, being 2 plus 3. 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), ... ( 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.

5 is the third Mersenne prime exponent of the form $2^n$ − $1$, which yields $31$: the prime index of the third Mersenne prime and second double Mersenne prime 127, as well as the third double Mersenne prime exponent for the number 2,147,483,647, which is the largest value that a signed 32-bit integer field can hold. There are only four known double Mersenne prime numbers, with a fifth candidate double Mersenne prime $M_$ = 2^{23058...93951} − 1 too large to compute with current computers. In a related sequence, the first 5 terms in the sequence of Catalan–Mersenne numbers $M_$ are the only known prime terms, with a sixth possible candidate in the order of 10^1037.7094. These prime sequences are conjectured to be prime up to a certain limit.

Every odd number greater than $1$ is the sum of at most five prime numbers, and every odd number greater than $5$ is conjectured to be expressible as the sum of three prime numbers. Helfgtott has provided a proof of the latter, also known as the odd Goldbach conjecture, that is already widely acknowledged by mathematicians as it still undergoes peer-review.

The sums of the first five non-primes greater than zero $1$ + $4$ + $6$ + $8$ + $9$ and the first five prime numbers $2$ + $3$ + $5$ + $7$ + $11$ both equal $28$; the 7th triangular number and like $6$ a perfect number, which also includes $496$, the 31st triangular number and perfect number of the form $2^$^−1($2^$ − $1$) with a $p$ of $5$, by the Euclid–Euler theorem.

There are a total of five known unitary perfect numbers, which are numbers that are the sums of their positive proper unitary divisors. A sixth unitary number, if discovered, would have at least nine odd prime factors.

Five is conjectured to be the only odd untouchable number, and if this is the case then five will be the only odd prime number that is not the base of an aliquot tree.

In figurate numbers, 5 is a pentagonal number