Natural number
23 (twenty-three ) is thenatural number following22 and preceding24 . It is aprime number .
Twenty-three is the ninthprime number , the smallest odd prime that is not atwin prime .[ 1] cousin prime with19 , and asexy prime with17 and29 ; while also being the largest member of the firstprime sextuplet (7 ,11 ,13 , 17, 19, 23).[ 2] Cunningham chain of the first kind (2 ,5 , 11, 23,47 ),[ 3] prime factors of the second set of consecutivediscrete semiprimes , (21 ,22 ). 23 is the smallest odd prime to be ahighly cototient number , as the solution tox − ϕ ( x ) {\displaystyle x-\phi (x)} 95 ,119 ,143 , and529 .[ 4]
23 is the secondSmarandache–Wellin prime inbase ten, as it is the concatenation of the decimal representations of the first two primes (2 and 3) and is itself also prime,[ 5] happy number .[ 6] The sum of the first nine primes up to 23 is asquare :2 + 3 + ⋯ + 23 = 100 = 10 2 {\displaystyle 2+3+\dots +23=100=10^{2}} 874 , which is divisible by 23, a property shared by few other numbers.[ 7] [ 8] It is the fifthfactorial prime ,[ 9] 14 , 23 is the firstPillai prime .[ 10] In the list offortunate numbers , 23 occurs twice, since adding 23 to either the fifth or eighthprimorial gives a prime number (namely 2333 and 9699713).[ 11] 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 is239 ). SeeWaring's problem . The twenty-thirdhighly composite number 20,160[ 12] super-prime 20,161) that cannot be expressed as the sum of twoabundant numbers .[ 13] Otherwise,46 = 23 × 2 {\displaystyle 46=23\times 2} even number that is not the sum of two abundant numbers. 23 is the secondWoodall prime ,[ 14] Eisenstein prime with noimaginary part andreal part of the form3 n − 1. {\displaystyle 3n-1.} Sophie Germain prime [ 15] safe prime .[ 16] 23 is the number oftrees on 8 unlabeled nodes.[ 17] Wedderburn–Etherington number , which are numbers that can be used to count certainbinary trees .[ 18] Thenatural logarithms of allpositive integers lower than 23 are known to have binaryBBP-type formulae .[ 19] 23 is the first primep for which unique factorization ofcyclotomic integers based on thep th root of unity breaks down.[ 20] 23 is the smallest positive solution toSunzi 's original formulation of theChinese remainder theorem . 23 is the smallestprime p {\displaystyle p} p {\displaystyle p} smooth numbers (11859210, 11859211) is the same as the largest consecutive pair of( p − 1 ) {\displaystyle (p-1)} [ 21] According to thebirthday 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 that365 times thenatural logarithm of 2, approximately 252.999, is very close to the number of pairs of 23 items and 22ndtriangular number ,253 . The first twenty-three odd prime numbers (between3 and89 inclusive), are allcluster primes p {\displaystyle p} k ≤ p − 3 {\displaystyle k\leq p-3} p {\displaystyle p} [ 23] 23 is the smallestdiscriminant of imaginary quadratic fields with class number 3 (negated),[ 24] complex cubic fields (also negated).[ 25] The twenty-thirdpermutable prime in decimalR 19 {\displaystyle R_{19}} prime repunit (afterR 2 {\displaystyle R_{2}} R 23 {\displaystyle R_{23}} R 1031 {\displaystyle R_{1031}} [ 26] [ 27] [ 28] [ 29] Hilbert's problems are twenty-three problems in mathematics published by German mathematician David Hilbert in 1900.
The firstMersenne number of the form2 n − 1 {\displaystyle 2^{n}-1} prime number when inputting a primeexponent is2047 = 23 × 89 , {\displaystyle 2047=23\times 89,} n = 11. {\displaystyle n=11.} [ 30]
On the other hand, the secondcomposite Mersenne number contains an exponentn {\displaystyle n} M 23 = 2 23 − 1 = 8 388 607 = 47 × 178 481 {\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 inbase ten :[ 31] [ 32] M 83 = 967...407 = 167 × 57 912 614 113 275 649 087 721 {\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,M 103 = 101 … 007 = 2 550 183 799 × 3 976 656 429 941 438 590 393 M 109 = 649 … 511 = 745 988 807 × 870 035 986 098 720 987 332 873 {\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 forM 23 {\displaystyle M_{23}} M 83 {\displaystyle M_{83}} 106 , which lies in between prime exponents ofM 103 {\displaystyle M_{103}} M 109 {\displaystyle M_{109}} 17 and18 ) in the sequence of Mersenne numbers sum to35 , which is the twenty-third composite number.[ 33]
23 ! {\displaystyle 23!} digits long in decimal, and there are only three other numbersn {\displaystyle n} factorials generate numbers that aren {\displaystyle n} 1 ,22 , and 24 .
TheLeech lattice Λ24 is a 24-dimensionallattice through which 23 otherpositive definite evenunimodular Niemeier lattices ofrank 24 are built, and vice-versa.Λ24 represents the solution to thekissing number in 24 dimensions as the precise lattice structure for the maximum number ofspheres that can fill 24-dimensional space without overlapping, equal to 196,560 spheres. These 23 Niemeier lattices are located atdeep holes ofradii √2 Conway group C 0 {\displaystyle \mathbb {C} _{0}}
By means of amatrix of the form( I a H / 2 H / 2 I b ) {\displaystyle \scriptstyle {\begin{pmatrix}Ia&H/2\\H/2&Ib\end{pmatrix}}} I {\displaystyle I} identity matrix andH {\displaystyle H} Hadamard matrix (Z /23Z ∪ ∞) witha = 2 andb = 3, andentries X(∞) = 1 and X(0) = -1 with X(n ) thequadratic residue symbolmod 23 for nonzeron . Through theextended binary Golay code B 24 {\displaystyle \mathbb {B} _{24}} Witt designW 24 {\displaystyle \mathbb {W} _{24}} , which produce a construction of the 196,560 minimalvectors in the Leech lattice. The extended binary Golay code is an extension of theperfect binary Golay code B 23 {\displaystyle \mathbb {B} _{23}} codewords ofsize 23.B 23 {\displaystyle \mathbb {B} _{23}} Mathieu groupM 23 {\displaystyle \mathbb {M} _{23}} as itsautomorphism group , which is the second largest member of thefirst generation in thehappy family ofsporadic groups .M 23 {\displaystyle \mathbb {M} _{23}} faithful complex representation in 22dimensions andgroup-3 actions on253 objects , with 253 equal to the number of pairs of objects in a set of 23 objects. In turn,M 23 {\displaystyle \mathbb {M} _{23}} Mathieu groupM 24 {\displaystyle \mathbb {M} _{24}} , which works throughW 24 {\displaystyle \mathbb {W} _{24}} 8 -elementoctads whose individual elements occur 253 times through its entireblock design . Using Niemer latticeD 24 ofgroup order 223 ·24! andCoxeter number 46 = 2·23, it can be made into amodule over thering of integers ofquadratic field Q ( − 23 ) {\displaystyle \mathbb {Q} ({\sqrt {-23}})} D 24 by anon-principal ideal of the ring of integers yields the Leech lattice. Conway andSloane provided constructions of the Leech lattice from all other 23 Niemeier lattices.[ 34]
Twenty-three four-dimensionalcrystal families exist within the classification ofspace groups . These are accompanied by sixenantiomorphic forms, maximizing the total count totwenty-nine crystal families.[ 35] cubes can be arranged to form twenty-threefree pentacubes , or twenty-nine distinctone-sided [ 36] [ 37]
There are 23 three-dimensionaluniform polyhedra that arecell facets insideuniform 4-polytopes that are not part of infinite families ofantiprismatic prisms andduoprisms : the fivePlatonic solids , the thirteenArchimedean solids , and five semiregularprisms (thetriangular ,pentagonal ,hexagonal ,octagonal , anddecagonal prisms).
23Coxeter groups ofparacompact hyperbolic honeycombs in thethird dimension generate151 uniqueWythoffian constructions of paracompact honeycombs. 23 four-dimensional Euclidean honeycombs are generated from theB ~ 4 {\displaystyle {\tilde {B}}_{4}} uniform polytopes are generated from theD 5 {\displaystyle \mathrm {D} _{5}}
Intwo-dimensional geometry, the regular 23-sidedicositrigon is the first regular polygon that is not constructible with acompass and straight edge or with the aide of anangle trisector (since it is neither aFermat prime nor aPierpont prime ), nor byneusis or a double-notched straight edge.[ 38] origami , however it is through other traditional methods for all regular polygons.[ 39]
Film and television [ edit ] In the TV seriesLost 6 reoccurring numbers (4, 8, 15, 16, 23, 42) that appear frequently throughout the show. ^ 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.)" .TheOn-Line Encyclopedia of Integer Sequences . OEIS Foundation. Retrieved5 December 2022 .^ Sloane, N. J. A. (ed.)."Sequence A001223 (Prime gaps: differences between consecutive primes.)" .TheOn-Line Encyclopedia of Integer Sequences . OEIS Foundation. Retrieved11 June 2023 .^ 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.)" .TheOn-Line Encyclopedia of Integer Sequences . OEIS Foundation. Retrieved11 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 primeA005384 ." ^ Sloane, N. J. A. (ed.)."Sequence A100827 (Highly cototient numbers)" .TheOn-Line Encyclopedia of Integer Sequences . OEIS Foundation. Retrieved31 May 2016 .^ Sloane, N. J. A. (ed.)."Sequence A069151 (Concatenations of consecutive primes, starting with 2, that are also prime)" .TheOn-Line Encyclopedia of Integer Sequences . OEIS Foundation. Retrieved31 May 2016 .^ Sloane, N. J. A. (ed.)."Sequence A007770 (Happy numbers)" .TheOn-Line Encyclopedia of Integer Sequences . OEIS Foundation. Retrieved31 May 2016 .^ (sequenceA045345 OEIS ) ^ "Puzzle 31.- The Average Prime number, APN(k) = S(Pk)/k" .www.primepuzzles.net . Retrieved29 November 2022 .^ Sloane, N. J. A. (ed.)."Sequence A088054 (Factorial primes)" .TheOn-Line Encyclopedia of Integer Sequences . OEIS Foundation. Retrieved31 May 2016 .^ Sloane, N. J. A. (ed.)."Sequence A063980 (Pillai primes)" .TheOn-Line Encyclopedia of Integer Sequences . OEIS Foundation. Retrieved31 May 2016 .^ Sloane, N. J. A. (ed.)."Sequence A005235 (Fortunate numbers)" .TheOn-Line Encyclopedia of Integer Sequences . OEIS Foundation. Retrieved31 May 2016 .^ 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.)" .TheOn-Line Encyclopedia of Integer Sequences . OEIS Foundation. Retrieved9 October 2023 .^ Sloane, N. J. A. (ed.)."Sequence A048242 (Numbers that are not the sum of two abundant numbers (not necessarily distinct).)" .TheOn-Line Encyclopedia of Integer Sequences . OEIS Foundation. Retrieved9 October 2023 .^ Sloane, N. J. A. (ed.)."Sequence A050918 (Woodall primes)" .TheOn-Line Encyclopedia of Integer Sequences . OEIS Foundation. Retrieved31 May 2016 .^ Sloane, N. J. A. (ed.)."Sequence A005384 (Sophie Germain primes)" .TheOn-Line Encyclopedia of Integer Sequences . OEIS Foundation. Retrieved31 May 2016 .^ Sloane, N. J. A. (ed.)."Sequence A005385 (Safe primes)" .TheOn-Line Encyclopedia of Integer Sequences . OEIS Foundation. Retrieved31 May 2016 .^ "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. Retrieved19 December 2021 .^ Sloane, N. J. A. (ed.)."Sequence A001190 (Wedderburn-Etherington numbers)" .TheOn-Line Encyclopedia of Integer Sequences . OEIS Foundation. Retrieved31 May 2016 .^ Chamberland, Marc."Binary BBP-Formulae for Logarithms and Generalized Gaussian-Mersenne Primes" (PDF) . ^ Weisstein, Eric W."Cyclotomic Integer" .mathworld.wolfram.com . Retrieved15 January 2019 . ^ Sloane, N. J. A. (ed.)."Sequence A228611 (Primes p such that the largest consecutive pair ofp {\displaystyle p} p − 1 {\displaystyle p-1} .TheOn-Line Encyclopedia of Integer Sequences . OEIS Foundation. Retrieved31 May 2016 .^ Weisstein, Eric W."Birthday Problem" .mathworld.wolfram.com . Retrieved19 August 2020 . ^ Sloane, N. J. A. (ed.)."Sequence A038133 (From a subtractive Goldbach conjecture: odd primes that are not cluster primes.)" .TheOn-Line Encyclopedia of Integer Sequences . OEIS Foundation. Retrieved26 December 2022 .^ Sloane, N. J. A. (ed.)."Sequence A006203 (Discriminants of imaginary quadratic fields with class number 3 (negated).)" .TheOn-Line Encyclopedia of Integer Sequences . OEIS Foundation. Retrieved20 March 2024 .^ Sloane, N. J. A. (ed.)."Sequence A023679 (Discriminants of complex cubic fields (negated).)" .TheOn-Line Encyclopedia of Integer Sequences . OEIS Foundation. Retrieved20 March 2024 .^ Guy, Richard;Unsolved Problems in Number Theory , p. 7ISBN 1475717385 ^ Sloane, N. J. A. (ed.)."Sequence A003459 (Absolute primes (or permutable primes): every permutation of the digits is a prime.)" .TheOn-Line Encyclopedia of Integer Sequences . OEIS Foundation. Retrieved10 January 2024 .^ Sloane, N. J. A. (ed.)."Sequence A004022 (Primes of the form (10^k - 1)/9. Also called repunit primes or repdigit primes.)" .TheOn-Line Encyclopedia of Integer Sequences . OEIS Foundation. Retrieved10 January 2024 .^ 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.)" .TheOn-Line Encyclopedia of Integer Sequences . OEIS Foundation. Retrieved10 January 2024 .^ Sloane, N. J. A. (ed.)."Sequence A000225 (Mersenne numbers)" .TheOn-Line Encyclopedia of Integer Sequences . OEIS Foundation. Retrieved16 February 2023 .^ Sloane, N. J. A. (ed.)."Sequence A136030 (Smallest prime factor of composite Mersenne numbers.)" .TheOn-Line Encyclopedia of Integer Sequences . OEIS Foundation. Retrieved12 June 2023 .^ Sloane, N. J. A. (ed.)."Sequence A136031 (Largest prime factor of composite Mersenne numbers.)" .TheOn-Line Encyclopedia of Integer Sequences . OEIS Foundation. Retrieved12 June 2023 .^ Sloane, N. J. A. (ed.)."Sequence A002808 (The composite numbers: numbers n of the form x*y for x > 1 and y > 1.)" .TheOn-Line Encyclopedia of Integer Sequences . OEIS Foundation. Retrieved9 January 2024 .^ 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 .^ Sloane, N. J. A. (ed.)."Sequence A004032 (Number of n-dimensional crystal families.)" .TheOn-Line Encyclopedia of Integer Sequences . OEIS Foundation. Retrieved21 November 2022 .^ Sloane, N. J. A. (ed.)."Sequence A000162 (Number of three dimensional polyominoes (or polycubes) with n cells.)" .TheOn-Line Encyclopedia of Integer Sequences . OEIS Foundation. Retrieved6 January 2023 .^ Sloane, N. J. A. (ed.)."Sequence A038119 (Number of n-celled solid polyominoes (or free polycubes, allowing mirror-image identification))" .TheOn-Line Encyclopedia of Integer Sequences . OEIS Foundation.^ 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 ^ P. Milici, R. DawsonThe equiangular compass December 1st, 2012, The Mathematical Intelligencer, Vol. 34, Issue 4https://www.researchgate.net/profile/Pietro_Milici2/publication/257393577_The_Equiangular_Compass/links/5d4c687da6fdcc370a8725e0/The-Equiangular-Compass.pdf ^ " .Christianity.com . Retrieved7 June 2021 .^ 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." ^ The Number 23 (2007) – Joel Schumacher | Synopsis, Characteristics, Moods, Themes and Related | AllMovie , retrieved12 August 2020
400 to 999
400s, 500s, and 600s 700s, 800s, and 900s
1000s and 10,000s
1000s 10,000s
100,000s to 10,000,000,000,000s
100,000 1,000,000 10,000,000 100,000,000 1,000,000,000 10,000,000,000 100,000,000,000 1,000,000,000,000 10,000,000,000,000
