Number 23

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 ${\displaystyle 3n-1.}$  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 ${\displaystyle x-\phi (x)}$  for the integers 95, 119, 143, and 529.

• 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.

• The sum of the first nine primes up to 23 is a square: ${\displaystyle 2+3+\dots +23=100=10^{2}}$  and 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.

• The twenty-third highly composite number 20,160 is one less than the last number (the 339th super-prime 20,161) that cannot be expressed as the sum of two abundant numbers.

Otherwise, ${\displaystyle 46=23\times 2}$  is the largest even number that is not the sum of two abundant numbers.

• 23 is the number of trees on 8 unlabeled nodes. It is also a Wedderburn–Etherington number, which are numbers that can be used to count certain binary trees.

• The natural logarithms of all positive integers lower than 23 are known to have binary BBP-type formulae.

• 23 is the first prime p for which unique factorization of cyclotomic integers based on the pth root of unity breaks down.

• 23 is the smallest positive solution to Sunzi's original formulation of the Chinese remainder theorem.

• 23 is the smallest prime ${\displaystyle p}$  such that the largest consecutive pair of ${\displaystyle p-}$ smooth numbers (11859210, 11859211) is the same as the largest consecutive pair of ${\displaystyle (p-1)-}$ 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 ${\displaystyle p}$  such that every even positive integer ${\displaystyle k\leq p-3}$  can be written as the sum of two prime numbers that do not exceed ${\displaystyle p}$ .

• 23 is the smallest discriminant of imaginary quadratic fields with class number 3 (negated), and it is the smallest discriminant of complex cubic fields (also negated).

• The twenty-third permutable prime in decimal ${\displaystyle R_{19}}$  is also the second to be a prime repunit (after ${\displaystyle R_{2}}$ ), followed by ${\displaystyle R_{23}}$  and ${\displaystyle R_{1031}}$ .

Hilbert's problems are twenty-three problems in mathematics published by German mathematician David Hilbert in 1900.

Mersenne numbers

The first Mersenne number of the form ${\displaystyle 2^{n}-1}$  that does not yield a prime number when inputting a prime exponent is ${\displaystyle 2047=23\times 89,}$  with ${\displaystyle n=11.}$ 

On the other hand, the second composite Mersenne number contains an exponent ${\displaystyle n}$  of twenty-three:

$${\displaystyle M_{23}=2^{23}-1=8\;388\;607=47\times 178\;481}$$

 

The twenty-third prime number (83) is an exponent to the fourteenth composite Mersenne number, which factorizes into two prime numbers, the largest of which is twenty-three digits long when written in base ten:

$${\displaystyle M_{83}=967...407=167\times 57\;912\;614\;113\;275\;649\;087\;721}$$

 

Further down in this sequence, the seventeenth and eighteenth composite Mersenne numbers have two prime factors each as well, where the largest of these are respectively twenty-two and twenty-four digits long,

$${\displaystyle {\begin{aligned}M_{103}&=101\ldots 007=2\;550\;183\;799\times 3\;976\;656\;429\;941\;438\;590\;393\\M_{109}&=649\ldots 511=745\;988\;807\times 870\;035\;986\;098\;720\;987\;332\;873\\\end{aligned}}}$$

 

Where prime exponents for ${\displaystyle M_{23}}$  and ${\displaystyle M_{83}}$  add to 106, which lies in between prime exponents of ${\displaystyle M_{103}}$  and ${\displaystyle M_{109}}$ , the index of the latter two (17 and 18) in the sequence of Mersenne numbers sum to 35, which is the twenty-third composite number.

${\displaystyle 23!}$  is twenty-three digits long in decimal, and there are only three other numbers ${\displaystyle n}$  whose factorials generate numbers that are ${\displaystyle n}$  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 ${\displaystyle \mathbb {C} _{0}}$ . The Leech lattice can be constructed in various ways, which include:

• By means of a matrix of the form ${\displaystyle \scriptstyle {\begin{pmatrix}Ia&H/2\\H/2&Ib\end{pmatrix}}}$  where ${\displaystyle I}$  is the identity matrix and ${\displaystyle H}$  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.

• Through the extended binary Golay code ${\displaystyle \mathbb {B} _{24}}$  and Witt design ${\displaystyle \mathbb {W} _{24}}$ , which produce a construction of the 196,560 minimal vectors in the Leech lattice. The extended binary Golay code is an extension of the perfect binary Golay code ${\displaystyle \mathbb {B} _{23}}$ , which has codewords of size 23. ${\displaystyle \mathbb {B} _{23}}$  has Mathieu group ${\displaystyle \mathbb {M} _{23}}$  as its automorphism group, which is the second largest member of the first generation in the happy family of sporadic groups. ${\displaystyle \mathbb {M} _{23}}$  has a minimum faithful complex representation in 22 dimensions and group-3 actions on 253 objects, with 253 equal to the number of pairs of objects in a set of 23 objects. In turn, ${\displaystyle \mathbb {M} _{23}}$  is the automorphism group of Mathieu group ${\displaystyle \mathbb {M} _{24}}$ , which works through ${\displaystyle \mathbb {W} _{24}}$  to generate 8-element octads whose individual elements occur 253 times through its entire block design.

• 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 ${\displaystyle \mathbb {Q} ({\sqrt {-23}})}$ , 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, maximizing the total count to twenty-nine crystal families. Five cubes can be arranged to form twenty-three free pentacubes, or twenty-nine distinct one-sided pentacubes (with 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, pentagonal, hexagonal, octagonal, and decagonal prisms).

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 ${\displaystyle {\tilde {B}}_{4}}$  cubic group, and 23 five-dimensional uniform polytopes are generated from the ${\displaystyle \mathrm {D} _{5}}$  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.

• The atomic number of vanadium.
• The atomic mass number of the stable isotope of sodium.
• Normal human sex cells have 23 chromosomes. Other human cells have 46 chromosomes, arranged in 23 pairs.
• Scientific notation for the Avogadro constant is written as 6.02214076×10²³.
• 23 is the width of the Arecibo message, sent to space in search for extraterrestrial intelligence.
• 23 is the TCP/IP port used for telnet and is the default for the telnet command.
• The Earth's axis is tilted at approximately 23°.

• 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.
• In I Ching divination, Hexagram 23 is 剝 (bō), "peeling" or "splitting apart".
• 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.

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, such as Neophyte Phenotype, Rock Paper Scissors, and Upside Down Bluejay, all of which have 23 tracks. His stage name also references the number.
• "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.

Film and television

• 23 is a German film about Karl Koch.
• In Jeepers Creepers, the Creeper appears every 23 years for 23 days to feast on human body parts
• In L: Change the World, the protagonist L signs his own name in the Death Note notebook and somehow knows that he has given himself 23 days to live, revealing a 23-day rule for the maximum number of days a person may live after they are added to the Japanese god of death's Death Note.
• The 1980s TV series Max Headroom was set at Network 23.
• In The Big Lebowski, the main characters deliberately use only lane 23 at the bowling alley.^{[citation needed]}
• In The Matrix Reloaded, the Architect tells Neo it is of utmost importance to choose 23 people to repopulate Zion.
• In the TV series Lost, 23 is one of the 6 reoccurring numbers (4, 8, 15, 16, 23, 42) that appear frequently throughout the show.
• The Number 23 is a 2007 film starring Jim Carrey about a man who becomes obsessed with the 23 enigma.

Other fields

• 23 skidoo (phrase) (sometimes 23 skiddoo) is an American slang phrase popularized during the early 20th century. 23 skidoo has been described as "perhaps the first truly national fad expression and one of the most popular fad expressions to appear in the U.S".
• The 23 enigma, proposed by William S. Burroughs plays a prominent role in the plot of the Illuminatus! Trilogy by Robert Shea and Robert Anton Wilson.
• The 23, in South Africa, refers to the 23 conscientious objectors who publicly refused to do military service in the Apartheid army in 1987. The following years the number increased to 143 (in 1988) and 771 (in 1989), with Apartheid being dismantled from 1990 onwards.
• X-23 is a character in the Marvel Universe. She is named for being the 23rd attempt to create a female genetic twin of Wolverine after attempts to create a male clone failed.
• 23 is the number of times Julius Caesar was stabbed in the Theatre of Pompey.

• Each national team competing in the FIFA Women's World Cup is allowed a 23-player squad. This squad size has been in place since 2015.