View source for 23 (number)

{{Use dmy dates|date=August 2020}}

{{Infobox number
| number =  23
| factorization = [[prime number|prime]]
| numeral = trivigesimal
| prime = 9th
| divisor = 1, 23
}}

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

== In mathematics ==
'''Twenty-three''' is the ninth [[prime number]], the smallest odd prime that is not a [[twin prime]].<ref>{{Cite OEIS |A007510 |Single (or isolated or non-twin) primes: Primes p such that neither p-2 nor p+2 is prime. |access-date=2022-12-05 }}</ref> It is, however, a [[cousin prime]] with [[19 (number)|19]], and a [[sexy prime]] with [[17 (number)|17]] and [[29 (number)|29]]; while also being the largest member of the first [[Prime quadruplet#Prime sextuplets|prime sextuplet]] ([[7]], [[11 (number)|11]], [[13 (number)|13]], 17, 19, 23).<ref>{{Cite OEIS |A001223 |Prime gaps: differences between consecutive primes. |access-date=2023-06-11 }}</ref> Twenty-three is also the next to last member of the first [[Cunningham chain]] of the first kind ([[2]], [[5]], 11, 23, [[47 (number)|47]]),<ref>{{Cite OEIS |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. |access-date=2023-06-11 }}
:"'''2, 5, 11, 23, 47''' is the complete Cunningham chain that begins with 2. Each term except the last is a Sophie Germain prime [[OEIS:A005384|A005384]]."</ref>  and the sum of the [[prime factor]]s of the second set of consecutive [[Semiprime#Examples and variations|discrete semiprimes]], ([[21 (number)|21]], [[22 (number)|22]]). 23 is the smallest odd prime to be a [[highly cototient number]], as the solution to <math>x-\phi(x)</math> for the integers [[95 (number)|95]], [[119 (number)|119]], [[143 (number)|143]], and [[529 (number)|529]].<ref>{{Cite OEIS|1=A100827|2=Highly cototient numbers|access-date=2016-05-31}}</ref>

* 23 is the second [[Smarandache–Wellin prime]] in [[Radix|base]] ten, as it is the concatenation of the decimal representations of the first two primes (2 and 3) and is itself also prime,<ref>{{Cite OEIS|1=A069151|2=Concatenations of consecutive primes, starting with 2, that are also prime|access-date=2016-05-31}}</ref> and a [[happy number]].<ref>{{Cite OEIS|1=A007770|2=Happy numbers|access-date=2016-05-31}}</ref>
* The sum of the first nine primes up to 23 is a [[Square number|square]]: <math>2 + 3 + \dots + 23 = 100 = 10^{2}</math> and the sum of the first 23 primes is [[874 (number)|874]], which is divisible by 23, a property shared by few other numbers.<ref>{{OEIS|id=A045345}}</ref><ref>{{Cite web|title=Puzzle 31.- The Average Prime number, APN(k) {{=}} S(Pk)/k|url=https://www.primepuzzles.net/puzzles/puzz_031.htm|access-date=2022-11-29|website=www.primepuzzles.net}}</ref>
* It is the fifth [[factorial prime]],<ref>{{Cite OEIS|1=A088054|2=Factorial primes|access-date=2016-05-31}}</ref> and since 14! + 1 is a multiple of 23, but 23 is not one more than a multiple of [[14 (number)|14]], 23 is the first [[Pillai prime]].<ref>{{Cite OEIS|1=A063980|2=Pillai primes|access-date=2016-05-31}}</ref>
* In the list of [[fortunate number]]s, 23 occurs twice, since adding 23 to either the fifth or eighth [[primorial]] gives a prime number (namely 2333 and 9699713).<ref>{{Cite OEIS|1=A005235|2=Fortunate numbers|access-date=2016-05-31}}</ref>
* 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 (number)|239]]). See [[Waring's problem]].
* The twenty-third [[highly composite number]] 20,160<ref>{{Cite OEIS |A002182 |Highly composite numbers, definition (1): numbers n where d(n), the number of divisors of n (A000005), increases to a record. |access-date=2023-10-09 }}</ref> is one less than the last number (the 339th [[super-prime]] 20,161) that cannot be expressed as the sum of two [[abundant number]]s.<ref>{{Cite OEIS |A048242 |Numbers that are not the sum of two abundant numbers (not necessarily distinct). |access-date=2023-10-09 }}</ref>
:Otherwise, <math>46 = 23 \times 2</math> is the largest [[Parity (mathematics)|even]] number that is not the sum of two abundant numbers.

* 23 is the second [[Woodall prime]],<ref>{{Cite OEIS|1=A050918|2=Woodall primes|access-date=2016-05-31}}</ref> and an [[Eisenstein prime]] with no [[imaginary part]] and [[real part]] of the form <math>3n-1.</math> It is the fifth [[Sophie Germain prime]]<ref>{{Cite OEIS|1=A005384|2=Sophie Germain primes|access-date=2016-05-31}}</ref> and the fourth [[safe prime]].<ref>{{Cite OEIS|1=A005385|2=Safe primes|access-date=2016-05-31}}</ref>
* 23 is the number of [[Tree (graph theory)|trees]] on 8 unlabeled nodes.<ref>{{Cite web|title=Sloane's A000055: Number of trees with n unlabeled nodes|url=https://oeis.org/A000055|url-status=live|access-date=2021-12-19|website=The On-Line Encyclopedia of Integer Sequences|publisher=OEIS Foundation|archive-url=https://web.archive.org/web/20101129012954/http://oeis.org:80/A000055 |archive-date=29 November 2010 }}</ref> It is also a [[Wedderburn–Etherington number]], which are numbers that can be used to count certain [[binary trees]].<ref>{{Cite OEIS|1=A001190|2=Wedderburn-Etherington numbers|access-date=2016-05-31}}</ref>
* The [[natural logarithm]]s of all [[positive integer]]s lower than 23 are known to have binary [[BBP-type formula]]e.<ref>{{Cite web |last1=Chamberland |first1=Marc |title=Binary BBP-Formulae for Logarithms and Generalized Gaussian-Mersenne Primes |url=https://chamberland.math.grinnell.edu/papers/bbp.pdf}}</ref>
* 23 is the first prime ''p'' for which unique factorization of [[cyclotomic field|cyclotomic integers]] based on the ''p''th root of unity breaks down.<ref>{{Cite web|url=http://mathworld.wolfram.com/CyclotomicInteger.html|title=Cyclotomic Integer|last=Weisstein|first=Eric W.|website=mathworld.wolfram.com|language=en|access-date=2019-01-15}}</ref>
* 23 is the smallest positive solution to [[Sunzi Suanjing|Sunzi]]'s original formulation of the [[Chinese remainder theorem]].
* 23 is the smallest [[prime number|prime]] <math>p</math> such that the largest consecutive pair of <math>p</math>-[[smooth numbers]] (11859210, 11859211) is the same as the largest consecutive pair of <math>(p-1)</math>-smooth numbers.<ref>{{Cite OEIS |A228611 |Primes p such that the largest consecutive pair of <math>p</math>-smooth integers is the same as the largest consecutive pair of <math>p-1</math>-smooth integers |access-date=2016-05-31 }}</ref>
* 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.<ref>{{Cite web|last=Weisstein|first=Eric W.|title=Birthday Problem|url=https://mathworld.wolfram.com/BirthdayProblem.html|access-date=2020-08-19|website=mathworld.wolfram.com|language=en}}</ref>{{pb}}A related coincidence is that [[365 (number)|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 (number)|253]].
* The first twenty-three odd prime numbers (between [[3]] and [[89 (number)|89]] inclusive), are all [[cluster prime]]s <math>p</math> such that every even positive integer <math>k \le p - 3</math> can be written as the sum of two prime numbers that do not exceed <math>p</math>.<ref>{{Cite OEIS |A038133 |From a subtractive Goldbach conjecture: odd primes that are not cluster primes. |access-date=2022-12-26 }}</ref>
* 23 is the smallest [[discriminant]] of imaginary quadratic fields with class number 3 (negated),<ref>{{Cite OEIS |A006203 |Discriminants of imaginary quadratic fields with class number 3 (negated). |access-date=2024-03-20 }}</ref> and it is the smallest discriminant of [[List of number fields with class number one#Cubic fields|complex cubic fields]] (also negated).<ref>{{Cite OEIS |A023679 |Discriminants of complex cubic fields (negated). |access-date=2024-03-20 }}</ref>
* The twenty-third [[permutable prime]] in decimal <math>R_{19}</math> is also the second to be a [[Repunit#Repunit primes|prime repunit]] (after <math>R_{2}</math>), followed by <math>R_{23}</math> and <math>R_{1031}</math>.<ref>Guy, Richard; ''Unsolved Problems in Number Theory'', p. 7 {{ISBN|1475717385}}</ref><ref>{{Cite OEIS |A003459 |Absolute primes (or permutable primes): every permutation of the digits is a prime. |access-date=2024-01-10}}</ref><ref>{{Cite OEIS |A004022 |Primes of the form (10^k - 1)/9. Also called repunit primes or repdigit primes. |access-date=2024-01-10 }}</ref><ref>{{Cite OEIS |A004023 |Indices of prime repunits: numbers n such that 11...111 (with n 1's) equal to (10^n - 1)/9 is prime. |access-date=2024-01-10 }}</ref>

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

=== Mersenne numbers ===

The first [[Mersenne Prime#Factorization of composite Mersenne numbers|Mersenne number]] of the form <math>2^{n} - 1</math> that does not yield a [[prime number]] when inputting a prime [[exponent]] is <math> 2047 = 23 \times 89,</math> with <math>n=11.</math><ref>{{Cite OEIS |A000225 |Mersenne numbers |access-date=2023-02-16 }}</ref>

On the other hand, the second [[Composite number|composite]] Mersenne number contains an exponent <math>n</math> of twenty-three:
<math display=block>M_{23} = 2^{23} - 1 = 8\;388\;607 = 47 \times 178\;481</math>

The twenty-third prime number ([[83 (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]]:<ref>{{Cite OEIS |A136030 |Smallest prime factor of composite Mersenne numbers. |access-date=2023-06-12 }}</ref><ref>{{Cite OEIS |A136031 |	Largest prime factor of composite Mersenne numbers. |access-date=2023-06-12 }}</ref>
<math display=block>M_{83} = 967...407 = 167 \times 57\;912\;614\;113\;275\;649\;087\;721</math>

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,
<math display=block>
\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}
</math>

Where prime exponents for <math>M_{23}</math> and <math>M_{83}</math> add to [[106 (number)|106]], which lies in between prime exponents of <math>M_{103}</math> and <math>M_{109}</math>, the index of the latter two ([[17 (number)|17]] and [[18 (number)|18]]) in the sequence of Mersenne numbers sum to [[35 (number)|35]], which is the twenty-third composite number.<ref>{{Cite OEIS |A002808 |The composite numbers: numbers n of the form x*y for x > 1 and y > 1. |access-date=2024-01-09 }}</ref>

<math>23!</math> is twenty-three [[Numerical digit|digits]] long in decimal, and there are only three other numbers <math>n</math> whose [[factorial]]s generate numbers that are <math>n</math> digits long in base ten: [[1]], [[22 (number)|22]], and&nbsp;[[24 (number)|24]].

=== In geometry ===

The '''[[Leech lattice]]''' '''Λ<sub>24</sub>''' is a 24-dimensional [[Lattice (group)|lattice]] through which 23 other [[Definite bilinear form|positive definite]] even [[Unimodular lattice|unimodular]] [[Niemeier lattice]]s of [[Rank of an abelian group|rank]] 24 are built, and vice-versa. '''Λ<sub>24</sub>''' represents the solution to the [[kissing number]] in 24 dimensions as the precise lattice structure for the maximum number of [[sphere]]s that can fill 24-dimensional space without overlapping, equal to 196,560 spheres. These 23 Niemeier lattices are located at ''deep holes'' of [[Radius|radii]] {{math|{{sqrt|2}}}} in lattice points around its automorphism group, [[Conway group]] <math>\mathbb C_{0}</math>. The Leech lattice can be constructed in various ways, which include:
* By means of a [[Matrix (mathematics)|matrix]] of the form <math>\scriptstyle\begin{pmatrix} Ia&H/2\\H/2&Ib\end{pmatrix}</math> where <math>I</math> is the [[identity matrix]] and <math>H</math> is a 24 by 24 [[Hadamard matrix]] ('''Z'''/23'''Z''' ∪ ∞) with ''a'' = 2 and ''b'' = 3, and [[Matrix (mathematics)#Definition|entries]] X(∞) = 1 and X(0) = -1 with X(''n'') the [[quadratic residue]] symbol [[Modulo operation|mod]] 23 for nonzero ''n''.
* Through the [[extended binary Golay code]] <math>\mathbb B_{24}</math> and [[Steiner system#The Steiner system S(5, 8, 24)|Witt design <math>\mathbb W_{24}</math>]], which produce a construction of the 196,560 minimal [[Row and column vectors|vectors]] in the Leech lattice. The extended binary Golay code is an extension of the [[Perfect code|perfect]] [[binary Golay code]] <math>\mathbb B_{23}</math>, which has [[Code word (communication)|codeword]]s of [[Block code#The alphabet Σ|size]]&nbsp;23. <math>\mathbb B_{23}</math> has [[Mathieu group M23|Mathieu group <math>\mathbb M_{23}</math>]] as its [[automorphism group]], which is the second largest member of the [[Sporadic group#Happy Family|first generation]] in the [[Sporadic group#Happy Family|happy family]] of [[sporadic groups]]. <math>\mathbb M_{23}</math> has a minimum [[Faithful representation|faithful complex representation]] in 22 [[dimension]]s and [[Group action|group-3 actions]] on [[253 (number)|253]] [[Group object|objects]], with 253 equal to the number of pairs of objects in a set of 23 objects. In turn, <math>\mathbb M_{23}</math> is the automorphism group of [[Mathieu group M24|Mathieu group <math>\mathbb M_{24}</math>]], which works through <math>\mathbb W_{24}</math> to generate '''8'''-element ''octads'' whose individual elements occur 253 times through its entire [[block design]].
* Using Niemer lattice '''D'''<sub>24</sub> of [[group order]] 2<sup>23</sup>·24! and [[Coxeter number]] 46 = 2·23, it can be made into a [[Module (mathematics)|module]] over the [[ring of integers]] of [[quadratic field]] <math>\mathbb{Q}(\sqrt{-23})</math>, whereby multiplying '''D'''<sub>24</sub> by a [[Principal ideal#Examples of non-principle ideals|non-principal ideal]] of the ring of integers yields the Leech lattice.

[[John Conway|Conway]] and [[Neil Sloane|Sloane]] provided constructions of the Leech lattice from all other 23 Niemeier lattices.<ref>{{Cite journal| last1=Conway | first1=John Horton | author1-link=John Horton Conway | last2=Sloane | first2=N. J. A. | author2-link=Neil Sloane | title=Twenty-three constructions for the Leech lattice | doi=10.1098/rspa.1982.0071 | mr=661720 | year=1982 | journal=[[Proceedings of the Royal Society A]] | issn=0080-4630 | volume=381 | issue=1781 | pages=275–283| bibcode=1982RSPSA.381..275C | s2cid=202575295 }}</ref>

Twenty-three four-dimensional [[Crystal system#Crystal family|crystal families]] exist within the classification of [[Space group#In other dimensions|space groups]]. These are accompanied by six [[Chirality (mathematics)|enantiomorphic]] forms, maximizing the total count to [[29 (number)|twenty-nine]] crystal families.<ref>{{Cite OEIS |A004032 |Number of n-dimensional crystal families. |access-date=2022-11-21 }}</ref> Five [[cube]]s can be arranged to form twenty-three [[Polycube#Enumerating polycubes|free pentacubes]], or twenty-nine distinct ''[[Polycubes#Enumerating polycubes|one-sided]]'' pentacubes (with reflections).<ref>{{Cite OEIS |A000162 |Number of three dimensional polyominoes (or polycubes) with n cells. |access-date=2023-01-06 }}</ref><ref>{{Cite OEIS |A038119 |Number of n-celled solid polyominoes (or free polycubes, allowing mirror-image identification) }}</ref>

There are 23 three-dimensional [[Uniform polyhedron|uniform polyhedra]] that are [[Facet (geometry)|cell facets]] inside [[uniform 4-polytope]]s that are not part of infinite families of [[Uniform antiprismatic prism|antiprismatic prisms]] and [[duoprisms]]: the five [[Platonic solid]]s, the thirteen [[Archimedean solid]]s, and five semiregular [[Prism (geometry)|prisms]] (the [[Triangular prism|triangular]], [[Pentagonal prism|pentagonal]], [[Hexagonal prism|hexagonal]], [[Octagonal prism|octagonal]], and [[Decagonal prism|decagonal]] prisms).

23 [[Coxeter group]]s of [[Paracompact honeycomb|paracompact hyperbolic honeycombs]] in the [[Three-dimensional space|third dimension]] generate [[151 (number)|151]] unique [[Wythoff construction|Wythoffian]] constructions of paracompact honeycombs. 23 four-dimensional Euclidean honeycombs are generated from the [[Uniform 5-polytope#Regular and uniform honeycombs|<math>{\tilde{B}}_4</math> cubic group]], and 23 five-dimensional [[Uniform 5-polytope#Enumerating the convex uniform 5-polytopes|uniform polytopes]] are generated from the [[D5 polytope|<math>\mathrm D_{5}</math> demihypercubic group]].

In [[Plane (geometry)|two-dimensional]] geometry, the regular 23-sided [[icositrigon]] is the first regular polygon that is not constructible with a [[Constructible polygon#Compass and straightedge constructions|compass and straight edge]] or with the aide of an [[angle trisection|angle trisector]] (since it is neither a [[Fermat prime]] nor a [[Pierpont prime]]), nor by [[neusis construction|neusis]] or a double-notched straight edge.<ref>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}}</ref> It is also not constructible with [[Mathematics of paper folding#Constructions|origami]], however it is through other traditional methods for all regular polygons.<ref>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</ref>

== 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.<ref>{{Cite web|title="The Lord is My Shepherd, I Shall Not Want" – Meaning of Psalm 23 Explained|url=https://www.christianity.com/wiki/bible/the-lord-is-my-shepherd-in-psalm-23.html|access-date=2021-06-07|website=Christianity.com|language=en}}</ref><ref>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."</ref>
* ''[[Principia Discordia]]'', the sacred text of [[Discordianism]], holds that 23 (along with the discordian prime [[5 (number)|5]]) is one of the sacred numbers of [[Eris (mythology)|Eris]], goddess of discord.

== In popular culture ==
=== Film and television ===
* In the TV series ''[[Lost (2004 TV series)|Lost]]'', 23 is one of the [[Mythology of Lost#The Numbers|6 reoccurring numbers]] (4, 8, 15, 16, 23, 42) that appear frequently throughout the show.

=== Other fields ===
{{See also|List of highways numbered 23}}
* [[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 Number 23]]'' is a 2007 film starring [[Jim Carrey]] about a man who becomes obsessed with the [[23 enigma]].<ref>{{Citation|title=The Number 23 (2007) – Joel Schumacher {{!}} Synopsis, Characteristics, Moods, Themes and Related {{!}} AllMovie|url=https://www.allmovie.com/movie/the-number-23-v336722|language=en-us|access-date=2020-08-12}}</ref> 
** 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.

== References ==
{{reflist|30em}}

== External links ==
{{Wikiquote}}
{{Commons category}}
* [https://web.archive.org/web/20141222234706/http://www.dauddegtre.co.uk/ 23 facts, 23 images, 23 gallery, and links to other 23's]

{{Integers|zero}}

[[Category:Integers]]