# 23 (number)

 ← 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.[1] It is, however, a cousin prime with 19, and a sexy prime with 17 as well as 29. Twenty-three is also the fifth factorial prime,[2] and the second Woodall prime.[3] It is an Eisenstein prime with no imaginary part and real part of the form 3n − 1. 23 is the fifth Sophie Germain prime[4] and the fourth safe prime,[5] 23 is 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.[6] 23 is the smallest odd prime to be a highly cototient number, as the solution to x − φ(x) for the integers 95, 119, 143, 529.[7] It is also a happy number in base-10.[8]

• 23 is the first prime p for which unique factorization of cyclotomic integers based on the p the root of unity breaks down.[10]
• The sum of the first 23 primes is 874, which is divisible by 23, a property shared by few other numbers.[11][12]
• 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).[13]
• 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.
• 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.[17] 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.
• 23 is the smallest prime number p such that the largest consecutive pair of p-smooth numbers is the same as the largest consecutive pair of (− 1)-smooth numbers.[18] That is, the largest consecutive pair of 23-smooth numbers (11859210, 11859211) is the same as the largest consecutive pair of 22-smooth numbers, where 23 is the smallest prime for which this is true.
• The first 23 odd prime numbers (up to 89) are all cluster primes p such that every even positive integer kp − 3 can be written as the sum of two prime numbers that do not exceed p.[19]
• The first Mersenne number of the form ${\displaystyle 2^{n}-1}$ that does not yield a prime number when inputting a prime exponent is 2047 (with ${\displaystyle n=11}$), which factorizes as 23 x 89.[20]
• 23! is twenty-three digits long in decimal. There are only three other numbers that have this property: 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.

Conway and Sloane provided constructions of the Leech lattice from all other 23 Niemeier lattices.[22]

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.[23] In three dimensions, five cubes can be arranged to form twenty-three free pentacubes, or twenty-nine distinct one-sided pentacubes (counting reflections).[24][25]

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 ${\displaystyle {\tilde {B}}_{4}}$ 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.[26] It is also not constructible with origami, however it is through other traditional methods for all regular polygons.[27]

## 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.[32][33] 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.[34][35]
• 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.[36]
• 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

• 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
• "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,[37] or because he first suffered an asthma attack on 23rd of the month.[38][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.[39]
• 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.[40]
• 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.
• 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.[41]

## 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.

## References

1. ^ Sloane, N. J. A. (ed.). "Sequence A007510 (Single (or isolated or non-twin) primes: Primes p such that neither p-2 nor p+2 is prime.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 5 December 2022.
2. ^ Sloane, N. J. A. (ed.). "Sequence A088054 (Factorial primes)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 31 May 2016.
3. ^ Sloane, N. J. A. (ed.). "Sequence A050918 (Woodall primes)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 31 May 2016.
4. ^ Sloane, N. J. A. (ed.). "Sequence A005384 (Sophie Germain primes)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 31 May 2016.
5. ^ Sloane, N. J. A. (ed.). "Sequence A005385 (Safe primes)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 31 May 2016.
6. ^ Sloane, N. J. A. (ed.). "Sequence A063980 (Pillai primes)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 31 May 2016.
7. ^ Sloane, N. J. A. (ed.). "Sequence A100827 (Highly cototient numbers)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 31 May 2016.
8. ^ Sloane, N. J. A. (ed.). "Sequence A007770 (Happy numbers)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 31 May 2016.
9. ^ Sloane, N. J. A. (ed.). "Sequence A069151 (Concatenations of consecutive primes, starting with 2, that are also prime)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 31 May 2016.
10. ^ Weisstein, Eric W. "Cyclotomic Integer". mathworld.wolfram.com. Retrieved 15 January 2019.
11. ^ (sequence A045345 in the OEIS)
12. ^ "Puzzle 31.- The Average Prime number, APN(k) = S(Pk)/k". www.primepuzzles.net. Retrieved 29 November 2022.
13. ^ Sloane, N. J. A. (ed.). "Sequence A005235 (Fortunate numbers)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 31 May 2016.
14. ^ "Sloane's A000055: Number of trees with n unlabeled nodes". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Archived from the original on 29 November 2010. Retrieved 19 December 2021.
15. ^ Sloane, N. J. A. (ed.). "Sequence A001190 (Wedderburn-Etherington numbers)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 31 May 2016.
16. ^ Chamberland, Marc. "Binary BBP-Formulae for Logarithms and Generalized Gaussian-Mersenne Primes" (PDF).
17. ^ Weisstein, Eric W. "Birthday Problem". mathworld.wolfram.com. Retrieved 19 August 2020.
18. ^ Sloane, N. J. A. (ed.). "Sequence A228611 (Primes p such that the largest consecutive pair of p-smooth integers is the same as the largest consecutive pair of p-1 smooth integers)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 31 May 2016.
19. ^ Sloane, N. J. A. (ed.). "Sequence A038133 (From a subtractive Goldbach conjecture: odd primes that are not cluster primes.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 26 December 2022.
20. ^ Sloane, N. J. A. (ed.). "Sequence A000225 (Mersenne numbers)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 16 February 2023.
21. ^ Guy, Richard; Unsolved Problems in Number Theory, p. 7 ISBN 1475717385
22. ^ Conway, John Horton; Sloane, N. J. A. (1982). "Twenty-three constructions for the Leech lattice". Proceedings of the Royal Society A. 381 (1781): 275–283. Bibcode:1982RSPSA.381..275C. doi:10.1098/rspa.1982.0071. ISSN 0080-4630. MR 0661720. S2CID 202575295.
23. ^ Sloane, N. J. A. (ed.). "Sequence A004032 (Number of n-dimensional crystal families.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 21 November 2022.
24. ^ Sloane, N. J. A. (ed.). "Sequence A000162 (Number of three dimensional polyominoes (or polycubes) with n cells.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 6 January 2023.
25. ^
26. ^ Arthur Baragar (2002) Constructions Using a Compass and Twice-Notched Straightedge, The American Mathematical Monthly, 109:2, 151-164, doi:10.1080/00029890.2002.11919848
27. ^ P. Milici, R. Dawson The equiangular compass December 1st, 2012, The Mathematical Intelligencer, Vol. 34, Issue 4 https://www.researchgate.net/profile/Pietro_Milici2/publication/257393577_The_Equiangular_Compass/links/5d4c687da6fdcc370a8725e0/The-Equiangular-Compass.pdf
28. ^ H. Wramsby, K. Fredga, P. Liedholm, "Chromosome analysis of human oocytes recovered from preovulatory follicles in stimulated cycles" New England Journal of Medicine 316 3 (1987): 121 – 124
29. ^ Barbara J. Trask, "Human genetics and disease: Human cytogenetics: 46 chromosomes, 46 years and counting" Nature Reviews Genetics 3 (2002): 769. "Human cytogenetics was born in 1956 with the fundamental, but empowering, discovery that normal human cells contain 46 chromosomes."
30. ^ Newell, David B.; Tiesinga, Eite (2019). The International System of Units (SI). NIST Special Publication 330. Gaithersburg, Maryland: National Institute of Standards and Technology. doi:10.6028/nist.sp.330-2019. S2CID 242934226.
31. ^ RFC 854, Telnet Protocol Specification
32. ^ ""The Lord is My Shepherd, I Shall Not Want" – Meaning of Psalm 23 Explained". Christianity.com. Retrieved 7 June 2021.
33. ^ Miriam Dunson, A Very Present Help: Psalm Studies for Older Adults. New York: Geneva Press (1999): 91. "Psalm 23 is perhaps the most familiar, the most loved, the most memorized, and the most quoted of all the psalms."
34. ^ Living Religions: An Encyclopaedia of the World's Faiths, Mary Pat Fisher, 1997, page 338, I.B. Tauris Publishers,
35. ^ Qur'an, Chapter 17, Verse 106
36. ^ Quran, Chapter 97
37. ^ Jarman, Douglas (1983). "Alban Berg, Wilhelm Fliess and the Secret Programme of the Violin Concerto". The Musical Times. 124 (1682): 218–223. doi:10.2307/962034. JSTOR 962034.
38. ^ Jarman, Douglas (1985). The Music of Alban Berg. University of California Press. ISBN 978-0-520-04954-3.
39. ^ 23 (1998) – Hans-Christian Schmid | Synopsis, Characteristics, Moods, Themes and Related | AllMovie, retrieved 12 August 2020
40. ^
41. ^ The Number 23 (2007) – Joel Schumacher | Synopsis, Characteristics, Moods, Themes and Related | AllMovie, retrieved 12 August 2020
42. ^ "Nan Cross: Supported men resisting apartheid conscription". Sunday Times. 22 July 2007. Retrieved 4 March 2023 – via PressReader.