← 22 23 24 →
Cardinaltwenty-three
Ordinal23rd
(twenty-third)
Numeral systemtrivigesimal
Factorizationprime
Prime9th
Divisors1, 23
Greek numeralΚΓ´
Roman numeralXXIII
Binary101112
Ternary2123
Senary356
Octal278
Duodecimal1B12

23 (twenty-three) is the natural number following 22 and preceding 24.

## In mathematics

Twenty-three is the ninth prime number, the smallest odd prime that is not a twin prime. It is, however, a cousin prime with 19, and a sexy prime with 17 and 29; while also being the largest member of the first prime sextuplet (7, 11, 13, 17, 19, 23). Twenty-three is also the fifth factorial prime, the second Woodall prime, and a happy number in decimal. It is an Eisenstein prime with no imaginary part and real part of the form $3n-1.$ It is also the fifth Sophie Germain prime and the fourth safe prime, and the next to last member of the first Cunningham chain of the first kind to have five terms (2, 5, 11, 23, 47). Since 14! + 1 is a multiple of 23, but 23 is not one more than a multiple of 14, 23 is the first Pillai prime. 23 is the smallest odd prime to be a highly cototient number, as the solution to $x-\phi (x)$ for the integers 95, 119, 143, and 529. The third decimal repunit prime after R2 and R19 is R23, followed by R1031.

• 23 is the second Smarandache–Wellin prime in base ten, as it is the concatenation of the decimal representations of the first two primes (2 and 3) and is itself also prime.
• It is the first prime p for which unique factorization of cyclotomic integers based on the pth root of unity breaks down.
• The sum of the first 23 primes is 874, which is divisible by 23, a property shared by few other numbers.
• In the list of fortunate numbers, 23 occurs twice, since adding 23 to either the fifth or eighth primorial gives a prime number (namely 2333 and 9699713).
• 23 has the distinction of being one of two integers that cannot be expressed as the sum of fewer than 9 cubes of positive integers (the other is 239). See Waring's problem.
• 23 is the smallest prime $p$ such that the largest consecutive pair of $p$ -smooth numbers (11859210, 11859211) is the same as the largest consecutive pair of ($p-1$ )-smooth numbers.
• According to the birthday paradox, in a group of 23 or more randomly chosen people, the probability is more than 50% that some pair of them will have the same birthday.
A related coincidence is that 365 times the natural logarithm of 2, approximately 252.999, is very close to the number of pairs of 23 items and 22nd triangular number, 253.
• The first twenty-three odd prime numbers (between 3 and 89 inclusive), are all cluster primes $p$ such that every even positive integer $k\leq p-3$ can be written as the sum of two prime numbers that do not exceed $p$ .

The first Mersenne number of the form $2^{n}-1$ that does not yield a prime number when inputting a prime exponent is $2047=23\times 89,$ with $n=11.$ On the other hand, the second composite Mersenne number contains an exponent $n$ of twenty-three:

$M_{23}=2^{23}-1=8\;388\;607=47\times 178\;481.$ Further in the sequence of Mersenne numbers, the 23rd prime number (83) is an exponent to the 14th composite Mersenne number, which factorizes into two prime numbers, the largest of which is twenty-three digits long when written in base ten:

$M_{83}=9\;671\;...\;649\;407=167\times 57\;912\;614\;113\;275\;649\;087\;721.$ $23!$ is also twenty-three digits long in decimal, and there are only three other numbers $n$ whose factorials generate numbers that are $n$ digits long in base ten: 1, 22, and 24.

### In geometry

The Leech lattice Λ24 is a 24-dimensional lattice through which 23 other positive definite even unimodular Niemeier lattices of rank 24 are built, and vice-versa. Λ24 represents the solution to the kissing number in 24 dimensions as the precise lattice structure for the maximum number of spheres that can fill 24-dimensional space without overlapping, equal to 196,560 spheres. These 23 Niemeier lattices are located at deep holes of radii 2 in lattice points around its automorphism group, Conway group $\mathbb {C} _{0)$ . The Leech lattice can be constructed in various ways, which include:

• By means of a matrix of the form ${\begin{pmatrix}Ia&H/2\\H/2&Ib\end{pmatrix))$ where $I$ is the identity matrix and $H$ is a 24 by 24 Hadamard matrix (Z/23Z ∪ ∞) with a = 2 and b = 3, and entries X(∞) = 1 and X(0) = -1 with X(n) the quadratic residue symbol mod 23 for nonzero n.
• Using Niemer lattice D24 of group order 223·24! and Coxeter number 46 = 2·23, it can be made into a module over the ring of integers of quadratic field $\mathbb {Q} ({\sqrt {-23)))$ , whereby multiplying D24 by a non-principal ideal of the ring of integers yields the Leech lattice.

Conway and Sloane provided constructions of the Leech lattice from all other 23 Niemeier lattices.

Twenty-three four-dimensional crystal families exist within the classification of space groups. These are accompanied by six enantiomorphic forms, which maximizes the total count to twenty-nine crystal families. In three dimensions, five cubes can be arranged to form twenty-three free pentacubes, or twenty-nine distinct one-sided pentacubes (counting reflections).

There are 23 three-dimensional uniform polyhedra that are cell facets inside uniform 4-polytopes that are not part of infinite families of antiprismatic prisms and duoprisms: the five Platonic solids, the thirteen Archimedean solids, and five semiregular prisms (the triangular prism, pentagonal prism, hexagonal prism, octagonal prism, and decagonal prism).

23 Coxeter groups of paracompact hyperbolic honeycombs in the third dimension generate 151 unique Wythoffian constructions of paracompact honeycombs. 23 four-dimensional Euclidean honeycombs are generated from the ${\tilde {B))_{4)$ cubic group, and 23 five-dimensional uniform polytopes are generated from the $\mathrm {D} _{5)$ demihypercubic group.

In two-dimensional geometry, the regular 23-sided icositrigon is the first regular polygon that is not constructible with a compass and straight edge or with the aide of an angle trisector (since it is neither a Fermat prime nor a Pierpont prime), nor by neusis or a double-notched straight edge. It is also not constructible with origami, however it is through other traditional methods for all regular polygons.

## In religion

• In Biblical numerology, it is associated with Psalm 23, also known as the Shepherd Psalm. It is possibly the most quoted and best known Psalm. Psalms is also the 23rd book in the Douay–Rheims Bible.
• In Islam, the Qur'an was revealed in a total of 23 years to Muhammed.
• Muslims believe the first verses of the Qur'an were revealed to the Islamic prophet Muhammad on the 23rd night of the 9th Islamic month, though, its disputed. 
• Principia Discordia, the sacred text of Discordianism, holds that 23 (along with the discordian prime 5) is one of the sacred numbers of Eris, goddess of discord.

## In popular culture

### Music

 Main article: 23 § Music
• Alfred Harth uses the number 23 in his artist name Alfred 23 Harth, or A23H, since the year 1+9+8+5 = 23.
• Twentythree is the name of Tristan Prettyman's debut album
• Twentythree an album by Carbon Based Lifeforms
• "Viginti Tres" (Latin for twenty-three) is a song by Tool on their album 10,000 Days
• Blink-182's song "What's My Age Again?" includes the lyrics "nobody likes you when you're 23."
• 23 is an album and title track by Blonde Redhead
• The Incubus song "Pardon Me" includes the lyrics "A decade ago, I never thought I would be, at 23, on the verge of spontaneous combustion, woe is me!" Frontman Brandon Boyd was 23 years old when he wrote the song and described himself as being "kind of obsessive about that number".
• "23" is a song by Jimmy Eat World, on their album Futures. The number also appears in the songs "Christmas Card" and "12."23".95" as well as on some items of clothing produced by the band.
• Four tet and Yellowcard both have songs titled "Twenty-Three".
• Dear 23, an album by The Posies
• Untitled 23, an album by The Church
• Noah23 has several albums which reference the number 23.[which?]
• "23 Minutes in Brussels", a song by Luna on their album Penthouse.
• The composer Alban Berg had a particular interest in the number 23, using it to structure several works. Various suggestions have been made as to the reason for this interest: that he took it from the Biorhythms theory of Wilhelm Fliess, in which a 23-day cycle is considered significant, or because he first suffered an asthma attack on 23rd of the month.[importance?]
• "23" is a single by Mike Will Made It
• On the cover of The Beatles' 1969 album Yellow Submarine the number 23 is displayed on the chest of one of the Blue Meanies.
• Network 23 refers to members of the Spiral Tribe. Sometimes 23 used to discretely mark the spots of a freetekno rave.
• The number 23 is used a lot throughout the visuals and music by the band Gorillaz, who have even devoted a whole page of their autobiography Rise Of The Ogre to the 23 enigma theory.

## In sports

• Each national team competing in the FIFA World Cup or FIFA Women's World Cup is allowed a 23-player squad. This squad size has been in place since 2002 for men and 2015 for women.
• Nissan typically uses this number for their Motorsport manufacturer teams, as the numbers 2 and 3 are pronounced "ni" and "san" in Japanese.
• 23 was basketball legend Michael Jordan's jersey number prior to his first retirement, then his chosen number again when he came out of retirement after a brief stint wearing the number 45.
• 23 was also the jersey number of Los Angeles Lakers small forward LeBron James, however he changed it to 6 in the 2021–22 NBA season.
• The maximum number of players on an NHL roster.
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