23 (number) explained

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 next to last member of the first Cunningham chain of the first kind (2, 5, 11, 23, 47),^[3] and the sum of the prime factors of the second set of consecutive discrete semiprimes, (21, 22). 23 is the smallest odd prime to be a highly cototient number, as the solution to

for the integers 95, 119, 143, and 529.^[4]

• 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,^[5] and a happy number.^[6]
• 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.^[7]

• It is the fifth factorial prime,^[8] and 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]
• 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).^[10]
• 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.
• The twenty-third highly composite number 20,160^[11] is one less than the last number (the 339th super-prime 20,161) that cannot be expressed as the sum of two abundant numbers.^[12]

Otherwise,

is the largest even number that is not the sum of two abundant numbers.

• 23 is the second Woodall prime,^[13] and an Eisenstein prime with no imaginary part and real part of the form

It is the fifth Sophie Germain prime^[14] and the fourth safe prime.^[15]

• 23 is the number of trees on 8 unlabeled nodes.^[16] It is also a Wedderburn–Etherington number, which are numbers that can be used to count certain binary trees.^[17]
• The natural logarithms of all positive integers lower than 23 are known to have binary BBP-type formulae.^[18]
• 23 is the first prime p for which unique factorization of cyclotomic integers based on the pth root of unity breaks down.^[19]
• 23 is the smallest positive solution to Sunzi's original formulation of the Chinese remainder theorem.

such that the largest consecutive pair of -smooth numbers (11859210, 11859211) is the same as the largest consecutive pair of -smooth numbers.^[20]

• 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.^[21] 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 .^[22]

• 23 is the smallest discriminant of imaginary quadratic fields with class number 3 (negated),^[23] and it is the smallest discriminant of complex cubic fields (also negated).^[24]
• The twenty-third permutable prime in decimal

is also the second to be a prime repunit (after ), followed by and .^{[25] [26] [27] [28]}

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 ^[29]

On the other hand, the second composite Mersenne number contains an exponent

of twenty-three:$$M\_\; =\; 2^\; -\; 1\; =\; 8\backslash ;388\backslash ;607\; =\; 47\; \backslash times\; 178\backslash ;481$$

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:^{[30] [31]} $$M\_\; =\; 967...407\; =\; 167\; \backslash times\; 57\backslash ;912\backslash ;614\backslash ;113\backslash ;275\backslash ;649\backslash ;087\backslash ;721$$

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,

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.^[32] 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 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. and Witt design , which produce a construction of the 196,560 minimal vectors in the Leech lattice. The extended binary Golay code is an extension of the perfect binary Golay code , which has codewords of size 23. has Mathieu group

M₂₃

as its automorphism group, which is the second largest member of the first generation in the happy family of sporadic groups. has a minimum faithful complex representation in 22 dimensions and group-3 actions on 253 objects, with 253 equal to the number of pairs of objects in a set of 23 objects. In turn, is the automorphism group of Mathieu group

M₂₄

, which works through to generate 8-element octads whose individual elements occur 253 times through its entire block design. , 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.^[33]

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

D₅

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

• The atomic number of vanadium.
• The atomic mass number of the stable isotope of sodium.
• Normal human sex cells have 23 chromosomes.^[38] Other human cells have 46 chromosomes, arranged in 23 pairs.^[39]
• In scientific notation, the Avogadro number is written as .^[40]

Notes and References

1. 2022-12-05 .
2. 2023-06-11 .
3. 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 A005384."

4. A100827. Highly cototient numbers. 2016-05-31.
5. A069151. Concatenations of consecutive primes, starting with 2, that are also prime. 2016-05-31.
6. A007770. Happy numbers. 2016-05-31.
7. Web site: Puzzle 31.- The Average Prime number, APN(k) S(Pk)/k. 2022-11-29. www.primepuzzles.net.
8. A088054. Factorial primes. 2016-05-31.
9. A063980. Pillai primes. 2016-05-31.
10. A005235. Fortunate numbers. 2016-05-31.
11. 2023-10-09 .
12. 2023-10-09 .
13. A050918. Woodall primes. 2016-05-31.
14. A005384. Sophie Germain primes. 2016-05-31.
15. A005385. Safe primes. 2016-05-31.
16. Web site: Sloane's A000055: Number of trees with n unlabeled nodes. live. 2021-12-19. The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. https://web.archive.org/web/20101129012954/http://oeis.org:80/A000055 . 29 November 2010 .
17. A001190. Wedderburn-Etherington numbers. 2016-05-31.
18. Web site: Chamberland . Marc . Binary BBP-Formulae for Logarithms and Generalized Gaussian-Mersenne Primes .
19. Web site: Cyclotomic Integer. Weisstein. Eric W.. mathworld.wolfram.com. en. 2019-01-15.
20. 2016-05-31 .
21. Web site: Weisstein. Eric W.. Birthday Problem. 2020-08-19. mathworld.wolfram.com. en.
22. 2022-12-26 .
23. 2024-03-20 .
24. 2024-03-20 .
25. Guy, Richard; Unsolved Problems in Number Theory, p. 7
26. 2024-01-10.
27. 2024-01-10 .
28. 2024-01-10 .
29. 2023-02-16 .
30. 2023-06-12 .
31. 2023-06-12 .
32. 2024-01-09 .
33. Conway . John Horton . John Horton Conway . Sloane . N. J. A. . Neil Sloane . Twenty-three constructions for the Leech lattice . 10.1098/rspa.1982.0071 . 661720 . 1982 . . 0080-4630 . 381 . 1781 . 275–283. 1982RSPSA.381..275C . 202575295 .
34. 2022-11-21 .
35. 2023-01-06 .
36. Arthur Baragar (2002) Constructions Using a Compass and Twice-Notched Straightedge, The American Mathematical Monthly, 109:2, 151-164,
37. 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
38. 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
39. 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."
40. Book: Newell . David B. . Tiesinga . Eite . 2019 . The International System of Units (SI) . NIST . NIST Special Publication 330 . National Institute of Standards and Technology . Gaithersburg, Maryland . 10.6028/nist.sp.330-2019 . 242934226 .