# 23 (number)

Short description: Natural number
 ← 22 23 24 →
Cardinaltwenty-three
Ordinal23rd
(twenty-third)
Numeral systemtrivigesimal
Factorizationprime
Prime9th
Divisors1, 23
Greek numeralΚΓ´
Roman numeralXXIII
Binary101112
Ternary2123
Quaternary1134
Quinary435
Senary356
Octal278
Duodecimal1B12
Vigesimal1320
Base 36N36

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 and 29; while also being the largest member of the first prime sextuplet (7, 11, 13, 17, 19, 23).[2] Twenty-three is also the fifth factorial prime,[3] the second Woodall prime,[4] and a happy number in decimal.[5] 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[6] and the fourth safe prime,[7] and the next to last member of the first Cunningham chain of the first kind to have five terms (2, 5, 11, 23, 47).[8] 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.[9] 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.[10]

• 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.[11]
• 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.[12][13]
• 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).[14]
• 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.
Otherwise, $\displaystyle{ 46 = 23 \times 2 }$ is the largest even number that is not the sum of two abundant numbers.
• 23 is the first prime p for which unique factorization of cyclotomic integers based on the pth root of unity breaks down.[20]
• 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.[21]
• 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.[22]
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 $\displaystyle{ k \le p - 3 }$ can be written as the sum of two prime numbers that do not exceed $\displaystyle{ p }$.[23]
• 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 $\displaystyle{ R_{1031} }$.[24][25][26][27]

### 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 $\displaystyle{ n=11. }$[28]

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:[29][30]

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

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{align} 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{align} }

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.[31]

$\displaystyle{ 23! }$ 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.
• 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.[32]

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.[33] Five cubes can be arranged to form twenty-three free pentacubes, or twenty-nine distinct one-sided pentacubes (with reflections).[34][35]

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.[36] It is also not constructible with origami, however it is through other traditional methods for all regular polygons.[37]

## 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.[42][43] 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.[44][45]
• 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. [46]
• 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
• 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".[47]
• "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,[48] or because he first suffered an asthma attack on 23rd of the month.[49][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.[50]
• 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.[51]
• 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.[52]

### 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.[53]
• 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.

## In sports

• 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.

## 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.)". OEIS Foundation. Retrieved 2022-12-05.
2. Sloane, N. J. A., ed. "Sequence A001223 (Prime gaps: differences between consecutive primes.)". OEIS Foundation. Retrieved 2023-06-11.
3. Sloane, N. J. A., ed. "Sequence A088054 (Factorial primes)". OEIS Foundation. Retrieved 2016-05-31.
4. Sloane, N. J. A., ed. "Sequence A050918 (Woodall primes)". OEIS Foundation. Retrieved 2016-05-31.
5. Sloane, N. J. A., ed. "Sequence A007770 (Happy numbers)". OEIS Foundation. Retrieved 2016-05-31.
6. Sloane, N. J. A., ed. "Sequence A005384 (Sophie Germain primes)". OEIS Foundation. Retrieved 2016-05-31.
7. Sloane, N. J. A., ed. "Sequence A005385 (Safe primes)". OEIS Foundation. Retrieved 2016-05-31.
8.
"2, 5, 11, 23, 47 is the complete Cunningham chain that begins with 2. Each term except the last is a Sophie Germain prime A005384."
9. Sloane, N. J. A., ed. "Sequence A063980 (Pillai primes)". OEIS Foundation. Retrieved 2016-05-31.
10. Sloane, N. J. A., ed. "Sequence A100827 (Highly cototient numbers)". OEIS Foundation. Retrieved 2016-05-31.
11. Sloane, N. J. A., ed. "Sequence A069151 (Concatenations of consecutive primes, starting with 2, that are also prime)". OEIS Foundation. Retrieved 2016-05-31.
12. (sequence A045345 in the OEIS)
13. Sloane, N. J. A., ed. "Sequence A005235 (Fortunate numbers)". OEIS Foundation. Retrieved 2016-05-31.
14. Sloane, N. J. A., ed. "Sequence A002182 (Highly composite numbers, definition (1): numbers n where d(n), the number of divisors of n (A000005), increases to a record.)". OEIS Foundation. Retrieved 2023-10-09.
15. Sloane, N. J. A., ed. "Sequence A048242 (Numbers that are not the sum of two abundant numbers (not necessarily distinct).)". OEIS Foundation. Retrieved 2023-10-09.
16. Sloane, N. J. A., ed. "Sequence A001190 (Wedderburn-Etherington numbers)". OEIS Foundation. Retrieved 2016-05-31.
17. Weisstein, Eric W.. "Cyclotomic Integer" (in en).
18. Weisstein, Eric W.. "Birthday Problem" (in en).
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20. Guy, Richard; Unsolved Problems in Number Theory, p. 7 ISBN:1475717385
21. Sloane, N. J. A., ed. "Sequence A003459 (Absolute primes (or permutable primes): every permutation of the digits is a prime.)". OEIS Foundation. Retrieved 2024-01-10.
22. Sloane, N. J. A., ed. "Sequence A004022 (Primes of the form (10^k - 1)/9. Also called repunit primes or repdigit primes.)". OEIS Foundation. Retrieved 2024-01-10.
23. Sloane, N. J. A., ed. "Sequence A004023 (Indices of prime repunits: numbers n such that 11...111 (with n 1's) equal to (10^n - 1)/9 is prime.)". OEIS Foundation. Retrieved 2024-01-10.
24. Sloane, N. J. A., ed. "Sequence A000225 (Mersenne numbers)". OEIS Foundation. Retrieved 2023-02-16.
25. Sloane, N. J. A., ed. "Sequence A136030 (Smallest prime factor of composite Mersenne numbers.)". OEIS Foundation. Retrieved 2023-06-12.
26. Sloane, N. J. A., ed. "Sequence A136031 (Largest prime factor of composite Mersenne numbers.)". OEIS Foundation. Retrieved 2023-06-12.
27. Sloane, N. J. A., ed. "Sequence A002808 (The composite numbers: numbers n of the form x*y for x > 1 and y > 1.)". OEIS Foundation. Retrieved 2024-01-09.
28. 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. doi:10.1098/rspa.1982.0071. ISSN 0080-4630. Bibcode1982RSPSA.381..275C.
29. Sloane, N. J. A., ed. "Sequence A004032 (Number of n-dimensional crystal families.)". OEIS Foundation. Retrieved 2022-11-21.
30. Sloane, N. J. A., ed. "Sequence A000162 (Number of three dimensional polyominoes (or polycubes) with n cells.)". OEIS Foundation. Retrieved 2023-01-06.
31. 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
32. 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
33. 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
34. 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."
35. 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.
36. RFC 854, Telnet Protocol Specification
37. 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."
38. Living Religions: An Encyclopaedia of the World's Faiths, Mary Pat Fisher, 1997, page 338, I.B. Tauris Publishers,
39. Qur'an, Chapter 17, Verse 106
40. Quran, Chapter 97
41. Rampton, Mike (October 19, 2019). "A Deep Dive Into Incubus' Pardon Me Video".
42. 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.
43. Jarman, Douglas (1985) (in en). The Music of Alban Berg. University of California Press. ISBN 978-0-520-04954-3.
44.