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From Wikipedia, the free encyclopedia

← 22  23  24 →
Cardinaltwenty-three
Ordinal23rd
(twenty-third)
Numeral systemtrivigesimal
Factorizationprime
Prime9th
Divisors1, 23
Greek numeralΚΓ´
Roman numeralXXIII
Binary101112
Ternary2123
Senary356
Octal278
Duodecimal1B12
Hexadecimal1716

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

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Transcription

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 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 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: 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, 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 such that the largest consecutive pair of smooth numbers (11859210, 11859211) is the same as the largest consecutive pair of 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 such that every even positive integer can be written as the sum of two prime numbers that do not exceed .[23]
  • The twenty-third permutable prime in decimal is also the second to be a prime repunit (after ), followed by and .[24][25][26][27]

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 that does not yield a prime number when inputting a prime exponent is with [28]

On the other hand, the second composite Mersenne number contains an exponent of twenty-three:

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]

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,

Where prime exponents for and add to 106, which lies in between prime exponents of and , 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]

is twenty-three digits long in decimal, and there are only three other numbers whose factorials generate numbers that are 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 . The Leech lattice can be constructed in various ways, which include:

  • By means of a matrix of the form where is the identity matrix and 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.[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  cubic group, and 23 five-dimensional uniform polytopes are generated from the  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 science and technology

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

Other fields

In sports

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 A001223 (Prime gaps: differences between consecutive primes.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 11 June 2023.
  3. ^ Sloane, N. J. A. (ed.). "Sequence A088054 (Factorial primes)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 31 May 2016.
  4. ^ Sloane, N. J. A. (ed.). "Sequence A050918 (Woodall primes)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 31 May 2016.
  5. ^ Sloane, N. J. A. (ed.). "Sequence A007770 (Happy numbers)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 31 May 2016.
  6. ^ Sloane, N. J. A. (ed.). "Sequence A005384 (Sophie Germain primes)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 31 May 2016.
  7. ^ Sloane, N. J. A. (ed.). "Sequence A005385 (Safe primes)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 31 May 2016.
  8. ^ Sloane, N. J. A. (ed.). "Sequence A192580 (Monotonic ordering of set S generated by these rules: if x and y are in S and xy+1 is a prime, then xy+1 is in S, and 2 is in S.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 11 June 2023.
    "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)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 31 May 2016.
  10. ^ Sloane, N. J. A. (ed.). "Sequence A100827 (Highly cototient numbers)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 31 May 2016.
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  15. ^ 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.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 9 October 2023.
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  18. ^ Sloane, N. J. A. (ed.). "Sequence A001190 (Wedderburn-Etherington numbers)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 31 May 2016.
  19. ^ Chamberland, Marc. "Binary BBP-Formulae for Logarithms and Generalized Gaussian-Mersenne Primes" (PDF).
  20. ^ Weisstein, Eric W. "Cyclotomic Integer". mathworld.wolfram.com. Retrieved 15 January 2019.
  21. ^ Sloane, N. J. A. (ed.). "Sequence A228611 (Primes p such that the largest consecutive pair of  -smooth integers is the same as the largest consecutive pair of -smooth integers)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 31 May 2016.
  22. ^ Weisstein, Eric W. "Birthday Problem". mathworld.wolfram.com. Retrieved 19 August 2020.
  23. ^ 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.
  24. ^ Guy, Richard; Unsolved Problems in Number Theory, p. 7 ISBN 1475717385
  25. ^ Sloane, N. J. A. (ed.). "Sequence A003459 (Absolute primes (or permutable primes): every permutation of the digits is a prime.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 10 January 2024.
  26. ^ Sloane, N. J. A. (ed.). "Sequence A004022 (Primes of the form (10^k - 1)/9. Also called repunit primes or repdigit primes.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 10 January 2024.
  27. ^ 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.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 10 January 2024.
  28. ^ Sloane, N. J. A. (ed.). "Sequence A000225 (Mersenne numbers)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 16 February 2023.
  29. ^ Sloane, N. J. A. (ed.). "Sequence A136030 (Smallest prime factor of composite Mersenne numbers.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 12 June 2023.
  30. ^ Sloane, N. J. A. (ed.). "Sequence A136031 (Largest prime factor of composite Mersenne numbers.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 12 June 2023.
  31. ^ Sloane, N. J. A. (ed.). "Sequence A002808 (The composite numbers: numbers n of the form x*y for x > 1 and y > 1.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 9 January 2024.
  32. ^ 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.
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  35. ^ Sloane, N. J. A. (ed.). "Sequence A038119 (Number of n-celled solid polyominoes (or free polycubes, allowing mirror-image identification))". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
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  44. ^ Living Religions: An Encyclopaedia of the World's Faiths, Mary Pat Fisher, 1997, page 338, I.B. Tauris Publishers,
  45. ^ Qur'an, Chapter 17, Verse 106
  46. ^ Quran, Chapter 97
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  48. ^ 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.
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External links

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