← 18 19 20 →
Cardinalnineteen
Ordinal19th
(nineteenth)
Factorizationprime
Prime8th
Divisors1, 19
Greek numeralΙΘ´
Roman numeralXIX
Binary100112
Ternary2013
Senary316
Octal238
Duodecimal1712
Hebrew numeralי"ט
Babylonian numeral𒌋𒐝

19 (nineteen) is the natural number following 18 and preceding 20. It is a prime number.

## Mathematics

${\displaystyle 19}$ is the eighth prime number, and forms a sexy prime with 13,[1] a twin prime with 17,[2] and a cousin prime with 23.[3] It is the third full reptend prime in decimal,[4] the fifth central trinomial coefficient,[5] and the seventh Mersenne prime exponent.[6] 19 is the second Keith number, and more specifically the first Keith prime.[7] It is also the second octahedral number, after 6.[8]

### Number theory

19 is the maximum number of fourth powers needed to sum up to any natural number, and in the context of Waring's problem, 19 is the fourth value of g(k).[9]

The Collatz sequence for nine requires nineteen steps to return back to one, more than any other number below it.[10] On the other hand, nineteen requires twenty steps, like eighteen. Less than ten thousand, only thirty-one other numbers require nineteen steps to return back to one:

{56, 58, 60, 61, 352, 360, 362, 368, 369, 372, 373, 401, 402, 403, 2176, ..., and 2421}.[11]

19 is the sixth Heegner number.[12] 67 and 163, respectively the 19th and 38th prime numbers, are the two largest Heegner numbers, of nine total.

#### Prime properties

The sum of the squares of the first 19 primes is divisible by 19.[13]

19 is the first prime number that is not a permutable prime in decimal, as its reverse (91) is composite; where 91 is also the fourth centered nonagonal number.[14]

1729 is also the nineteenth dodecagonal number.[17]

19, alongside 109, 1009, and 10009, are all prime (with 109 also full reptend), and form part of a sequence of numbers where inserting a digit inside the previous term produces the next smallest prime possible, up to scale, with the composite number 9 as root.[18] 100019 is the next such smallest prime number, by the insertion of a 1.

• Numbers of the form 10n9 equivalent to 10x + 9 with x = n + 1, where n is the number of zeros in the term, are prime for n = {0, 1, 2, 3, 8, 17, 21, 44, 48, 55, 68, 145, 201, 271, 2731, 4563}, and probably prime for n = {31811, 43187, 48109, 92691}.[19]

R19 is the second base-10 repunit prime, short for the number 1111111111111111111.[20]

#### Figurate numbers and magic figures

19 is the third centered triangular number as well as the third centered hexagonal number.[21][22]

19 is the first number in an infinite sequence of numbers in decimal whose digits start with 1 and have trailing 9's, that form triangular numbers containing trailing zeroes in proportion to 9s present in the original number; i.e. 19900 is the 199th triangular number, and 1999000 is the 1999th.[24]
• Like 19, 199 and 1999 are also both prime, as are 199999 and 19999999. In fact, a number of the form 19n, where n is the number of nines that terminate in the number, is prime for:
n = {1, 2, 3, 5, 7, 26, 27, 53, 147, 236, 248, 386, 401}.[25]

The number of nodes in regular hexagon with all diagonals drawn is nineteen.[26]

• Distinguishably, the only nontrivial normal magic hexagon is composed of nineteen cells, where every diagonal of consecutive hexagons has sums equal to 38, or twice 19.[27]
• A hexaflexagon is a strip of nineteen alternating triangular faces that can flex into a regular hexagon, such that any two of six colorings on triangles can be oriented to align on opposite sides of the folded figure.[28]
• Nineteen is also the number of one-sided hexiamonds, meaning there are nineteen ways of arranging six equiangular triangular polyforms edge-to-edge on the plane without turn-overs (and where holes are allowed).[29]

${\displaystyle {\tfrac {1}{19))}$ can be used to generate the first full, non-normal prime reciprocal magic square in decimal whose rows, columns and diagonals — in a 18 x 18 array — all generate a magic constant of 81 = 92.[30]

• The next prime number to generate a like-magic square in base-ten is 383,[31] the seventy-sixth prime number (where 19 × 4 = 76).[32] A regular 19 x 19 magic square, on the other hand, has a magic constant ${\displaystyle M_{19))$ of 3439 = 19 × 181.[33]

### In abstract algebra

The projective special linear group ${\displaystyle \mathrm {L(19)} }$ represents the abstract structure of the 57-cell: a universal 4-polytope with a total of one hundred and seventy-one (171 = 9 × 19) edges and vertices, and fifty-seven (57 = 3 × 19) hemi-icosahedral cells that are self-dual.[34]

In total, there are nineteen Coxeter groups of non-prismatic uniform honeycombs in the fourth dimension: five Coxeter honeycomb groups exist in Euclidean space, while the other fourteen Coxeter groups are compact and paracompact hyperbolic honeycomb groups.

• There are also specifically nineteen uniform honeycombs inside the Euclidean ${\displaystyle {\tilde {C))_{4))$ tesseractic honeycomb group in 4-space. In 5-space, there are nineteen uniform polytopes with ${\displaystyle \mathrm {A} _{5))$ simplex symmetry.

There are infinitely many finite-volume Vinberg polytopes up through dimension nineteen, which generate hyperbolic tilings with degenerate simplex quadrilateral pyramidal domains, as well as prismatic domains and otherwise.[35]

On the other hand, a cubic surface is the zero set in ${\displaystyle \mathbb {P^{3)) }$ of a homogeneous cubic polynomial in four variables ${\displaystyle f=c_{3}000x_{1}^{3}+c_{2}100x_{1}^{2}x_{2}+c_{1}200x^{1}x_{2}^{2}+c_{0}300x_{2}^{3}+\cdots +c_{0}003x_{4}^{3},}$ a polynomial with a total of twenty coefficients, which specifies a space for cubic surfaces that is 19-dimensional.[37]

#### Finite simple groups

19 is the eighth consecutive supersingular prime. It is the middle indexed member in the sequence of fifteen such primes that divide the order of the Friendly Giant ${\displaystyle \mathrm {F_{1)) }$, the largest sporadic group: {2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 41, 47, 59, 71}.[38]

• Janko groups ${\displaystyle \mathrm {J_{1)) }$ and ${\displaystyle \mathrm {J_{3)) }$ are the two-smallest of six pariah groups that are not subquotients of ${\displaystyle \mathrm {F_{1)) }$, which contain 19 as the largest prime number that divides their orders.[39]
${\displaystyle \mathrm {J_{1)) }$ holds (2,3,7) as standard generators (a,b,ab) that yield a semi-presentation where o(abab2) = 19, while ${\displaystyle \mathrm {J_{3)) }$ holds as standard generators (2A, 3A, 19), where o([a, b]) = 9.[40][41]
• ${\displaystyle 10944}$ is the dimensionality of the minimal faithful complex representation of O'Nan group ${\displaystyle O'N}$ — the second-largest after ${\displaystyle \mathrm {F_{1)) }$ of like-representation in ${\displaystyle {\text{dim)){\text{ ))196,883}$ and largest amongst the six pariahs[42] — whose value lies midway between primes (10939, 10949), the latter with a prime index of ${\displaystyle 1330}$,[43] which is the nineteenth tetrahedral number.[44]
• On the other hand, the Tits group ${\displaystyle \mathrm {T} }$, as the only non-strict group of Lie type that can loosely categorize as sporadic, has group order 211 · 33 · 52 · 13, whose prime factors (inclusive of powers) generate a sum equal to 54, which is the smallest non-trivial 19-gonal number.[45]

In the Happy Family of sporadic groups, nineteen of twenty-six such groups are subquotients of the Friendly Giant, which is also its own subquotient.[46] If the Tits group is indeed included as a group of Lie type,[47] then there are nineteen classes of finite simple groups that are not sporadic groups.

Worth noting, 26 is the only number to lie between a perfect square (52) and a cube (33); if all primes in the prime factorizations of 25 and 27 are added together, a sum of 19 is obtained.

## Religion

### Islam

• The number of angels guarding Hell ("Hellfire") ("Saqar") according to the Qur'an: "Over it are nineteen" (74:30), after which the Qur'an describes this number as being "a trial for those who disbelieve" (74:31), a sign for people of the scripture to be "convinced" (74:31) and that believers "will increase in faith" (74:31) due to it.
• The Number of Verse and Sura together in the Qur'an which announces Jesus son of Maryam's (Mary's) birth (Qur'an 19:19).
• A group called United Submitter International claim the Quran has mathematical structure based on the number 19. The gematrical value of WAHD = 6+1+8+4=19, Wahd means 'One' (God) to the first verse (1:1), known as Bas-malah, consists of 19 Arabic letters or the Quran consists of 114 (19x6) surat etc.

### Baháʼí faith

In the Bábí and Baháʼí Faiths, a group of 19 is called a Váhid, a Unity (Arabic: واحد, romanizedwāhid, lit.'one'). The numerical value of this word in the Abjad numeral system is 19.

### Celtic paganism

19 is a sacred number of the goddess Brigid because it is said to represent the 19-year cycle of the Great Celtic Year and the amount of time it takes the Moon to coincide with the winter solstice.[48]

## Music

• "19" is a 1985 song by Paul Hardcastle, including sampled soundbites taken from a documentary about the Vietnam War in which 19 is claimed to have been the average age of United States soldiers killed in the conflict.[49] The song was parodied by British satirist Rory Bremner under the pseudonym 'The Commentators,' as N-n-nineteen, Not Out, the title referring to the batting average of David Gower, the England cricket captain, during his side's risible performance against the West Indies in 1984 when they lost 5–0.
• "I Was Only Nineteen" by the Australian group Redgum reached number one on the Australian charts in 1983. In 2005 a hip hop version of the song was produced by The Herd.
• 19 is the name of Adele's 2008 debut album, so named since she was 19 years old at the time.
• "Hey Nineteen" is a song by American jazz rock band Steely Dan, on the 1980 album Gaucho.
• Nineteen has been used as an alternative to twelve for a division of the octave into equal parts. This idea goes back to Salinas in the sixteenth century, and is interesting in part because it gives a system of meantone tuning, being close to 1/3 comma meantone. See 19 equal temperament.
• Some organs use the 19th harmonic to approximate a minor third.

## Games

• The game of Go is played on a grid of 19×19 lines (though variants can be played on grids of other sizes).
• Though the maximum score for a cribbage hand is 29, there is no combination of cards that adds up to 19 points. Many cribbage players, therefore, jokingly refer to a zero-point hand as "a 19 hand".
• In the base version of Settlers of Catan there are 19 hexagonal pieces that can be randomly or intentionally placed to form the board.

## In sports

• In golf, the '19th hole' is the clubhouse bar and in match play, if there is a tie after 18 holes, an extra hole(s) is played. In miniature golf it is an extra hole on which the winner earns an instant prize.

## References

1. ^ Sloane, N. J. A. (ed.). "Sequence A046117 (Primes p such that p-6 is also prime. (Upper of a pair of sexy primes.))". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2022-08-05.
2. ^ Sloane, N. J. A. (ed.). "Sequence A006512 (Greater of twin primes.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2022-08-05.
3. ^ Sloane, N. J. A. (ed.). "Sequence A088762 (Numbers n such that (2n-1, 2n+3) is a cousin prime pair.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2022-08-05.
4. ^ Sloane, N. J. A. (ed.). "Sequence A001913 (Full reptend primes: primes with primitive root 10.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2022-08-05.
5. ^ Sloane, N. J. A. (ed.). "Sequence A002426 (Central trinomial coefficients: largest coefficient of (1 + x + x^2)^n.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2022-08-05.
6. ^ "Sloane's A000043 : Mersenne exponents". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2016-05-31.
7. ^ Sloane, N. J. A. (ed.). "Sequence A007629 (Repfigit (REPetitive FIbonacci-like diGIT) numbers (or Keith numbers).)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2022-08-05.
8. ^ Sloane, N. J. A. (ed.). "Sequence A005900 (Octahedral numbers)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2016-08-17.
9. ^ Sloane, N. J. A. (ed.). "Sequence A002804 ((Presumed) solution to Waring's problem.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2022-08-05.
10. ^ Sloane, N. J. A. "3x+1 problem". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-01-24.
11. ^ Sloane, N. J. A. (ed.). "Sequence A006577 (Number of halving and tripling steps to reach 1 in '3x+1' problem, or -1 if 1 is never reached)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-01-24.
"Table of n, a(n) for n = 1..10000".
12. ^ "Sloane's A003173 : Heegner numbers". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2016-05-31.
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14. ^ a b Sloane, N. J. A. (ed.). "Sequence A060544 (Centered 9-gonal (also known as nonagonal or enneagonal) numbers. Every third triangular number, starting with 1)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2022-11-30.
15. ^ "19". Prime Curios!. Retrieved 2022-08-05.
16. ^ Sloane, N. J. A. (ed.). "Sequence A005349 (Niven (or Harshad, or harshad) numbers: numbers that are divisible by the sum of their digits.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2022-10-11.
17. ^ Sloane, N. J. A. (ed.). "Sequence A051624 (12-gonal (or dodecagonal) numbers: a(n) equal to n*(5*n-4).)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-12-21.
18. ^
19. ^ Sloane, N. J. A. (ed.). "Sequence A088275 (Numbers n such that 10^n + 9 is prime)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2022-07-28.
20. ^ Guy, Richard; Unsolved Problems in Number Theory, p. 7 ISBN 1475717385
21. ^ "Sloane's A125602 : Centered triangular numbers that are prime". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2016-05-31.
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23. ^ Sloane, N. J. A. (ed.). "Sequence A000217 (Triangular numbers)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2022-07-13.
24. ^ Sloane, N. J. A. "Sequence A186076". The On-line Encyclopedia of Integer Sequences. Retrieved 2022-07-13. Note that terms A186074(4) and A186074(10) have trailing 0's, i.e. 19900 = Sum_{k=0..199} k and 1999000 = Sum_{k=0..1999} k...". "This pattern continues indefinitely: 199990000, 19999900000, etc.
25. ^ Sloane, N. J. A. (ed.). "Sequence A055558 (Primes of the form 1999...999)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2022-07-26.
26. ^ Sloane, N. J. A. (ed.). "Sequence A007569 (Number of nodes in regular n-gon with all diagonals drawn.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-04-04.
27. ^ Trigg, C. W. (February 1964). "A Unique Magic Hexagon". Recreational Mathematics Magazine. Retrieved 2022-07-14.
28. ^ Gardner, Martin (January 2012). "Hexaflexagons". The College Mathematics Journal. 43 (1). Taylor & Francis: 2–5. doi:10.4169/college.math.j.43.1.002. JSTOR 10.4169/college.math.j.43.1.002. S2CID 218544330.
29. ^ Sloane, N. J. A. (ed.). "Sequence A006534 (Number of one-sided triangular polyominoes (n-iamonds) with n cells; turning over not allowed, holes are allowed.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-12-08.
30. ^ Andrews, William Symes (1917). Magic Squares and Cubes (PDF). Chicago, IL: Open Court Publishing Company. pp. 176, 177. ISBN 9780486206585. MR 0114763. OCLC 1136401. Zbl 1003.05500.
31. ^
32. ^ Sloane, N. J. A. (ed.). "Sequence A000040 (The prime numbers)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-09-06.
33. ^ Sloane, N. J. A. (ed.). "Sequence A006003 (a(n) equal to n*(n^2 + 1)/2.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-09-04.
34. ^ Coxeter, H. S. M. (1982). "Ten toroids and fifty-seven hemidodecahedra". Geometriae Dedicata. 13 (1): 87–99. doi:10.1007/BF00149428. MR 0679218. S2CID 120672023.
35. ^ Allcock, Daniel (11 July 2006). "Infinitely many hyperbolic Coxeter groups through dimension 19". Geometry & Topology. 10 (2): 737–758. arXiv:0903.0138. doi:10.2140/gt.2006.10.737. S2CID 14378861.
36. ^ Tumarkin, P. (2004). "Hyperbolic Coxeter n-polytopes with n + 2 facets". Mathematical Notes. 75 (5/6). Springer: 848–854. arXiv:math/0301133v2. doi:10.1023/B:MATN.0000030993.74338.dd. MR 2086616. S2CID 15156852. Zbl 1062.52012.
37. ^ Seigal, Anna (2020). "Ranks and symmetric ranks of cubic surfaces". Journal of Symbolic Computation. 101. Amsterdam: Elsevier: 304–306. arXiv:1801.05377. Bibcode:2018arXiv180105377S. doi:10.1016/j.jsc.2019.10.001. S2CID 55542435. Zbl 1444.14091.
38. ^ Sloane, N. J. A. (ed.). "Sequence A002267 (The 15 supersingular primes.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2022-12-11.
39. ^ Ronan, Mark (2006). Symmetry and the Monster: One of the Greatest Quests of Mathematics. New York: Oxford University Press. pp. 244–246. doi:10.1007/s00283-008-9007-9. ISBN 978-0-19-280722-9. MR 2215662. OCLC 180766312. Zbl 1113.00002.
40. ^ Wilson, R.A (1998). "Chapter: An Atlas of Sporadic Group Representations" (PDF). The Atlas of Finite Groups - Ten Years On (LMS Lecture Note Series 249). Cambridge, U.K: Cambridge University Press. p. 267. doi:10.1017/CBO9780511565830.024. ISBN 9780511565830. OCLC 726827806. S2CID 59394831. Zbl 0914.20016.
List of standard generators of all sporadic groups.
41. ^ Nickerson, S.J.; Wilson, R.A. (2011). "Semi-Presentations for the Sporadic Simple Groups". Experimental Mathematics. 14 (3). Oxfordshire: Taylor & Francis: 365. CiteSeerX 10.1.1.218.8035. doi:10.1080/10586458.2005.10128927. MR 2172713. S2CID 13100616. Zbl 1087.20025.
42. ^
43. ^ Sloane, N. J. A. (ed.). "Sequence A000040 (The prime numbers.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2024-02-28.
44. ^ Sloane, N. J. A. (ed.). "Sequence A000292". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2024-02-28.
45. ^ Sloane, N. J. A. (ed.). "Sequence A051871 (19-gonal (or enneadecagonal) numbers: n(17n-15)/2.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-12-09.
46. ^ John F.R. Duncan; Michael H. Mertens; Ken Ono (2017). "Pariah moonshine". Nature Communications. 8 (1): 2 (Article 670). arXiv:1709.08867. Bibcode:2017NatCo...8..670D. doi:10.1038/s41467-017-00660-y. PMC 5608900. PMID 28935903. ...so [sic] moonshine illuminates a physical origin for the monster, and for the 19 other sporadic groups that are involved in the monster.
47. ^ R. B. Howlett; L. J. Rylands; D. E. Taylor (2001). "Matrix generators for exceptional groups of Lie type". Journal of Symbolic Computation. 31 (4): 429. doi:10.1006/jsco.2000.0431. ...for all groups of Lie type, including the twisted groups of Steinberg, Suzuki and Ree (and the Tits group).
48. ^ Brigid: Triple Goddess of the Flame (Health, Hearth, & Forge)
49. ^ Roush, Gary (2008-06-02). "Statistics about the Vietnam War". Vietnam Helicopter Flight Crew Network. Archived from the original on 2010-01-06. Retrieved 2009-12-06. Assuming KIAs accurately represented age groups serving in Vietnam, the average age of an infantryman (MOS 11B) serving in Vietnam to be 19 years old is a myth, it is actually 22. None of the enlisted grades have an average age of less than 20.