19 (number)

Source: Wikipedia, the free encyclopedia.
← 18 19 20 →
nonadecimal
Factorizationprime
Prime8th
Divisors1, 19
Greek numeralΙΘ´
Roman numeralXIX
Binary100112
Ternary2013
Senary316
Octal238
Duodecimal1712
Hexadecimal1316
Hebrew numeralי"ט
Babylonian numeral𒌋𒐝

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

Mathematics

19 is a centered triangular number.

is the eighth

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

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

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

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.

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
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

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
    plane without turn-overs (and where holes are allowed).[29]

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]

In abstract algebra

The

projective special linear group
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 tesseractic honeycomb group in 4-space. In 5-space, there are nineteen uniform polytopes with 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 of a homogeneous

cubic polynomial
in four variables 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

indexed member in the sequence of fifteen such primes that divide the order of the Friendly Giant
, the largest sporadic group: {2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 41, 47, 59, 71}.[38]

  • Janko groups and are the two-smallest of six pariah groups that are not subquotients of , which contain 19 as the largest prime number that divides their orders.[39]
holds (2,3,7) as standard generators (a,b,ab) that yield a semi-presentation where o(abab2) = 19, while holds as standard generators (2A, 3A, 19), where o([a, b]) = 9.[40][41]
  • is the dimensionality of the minimal faithful complex representation of O'Nan group — the second-largest after of like-representation in and largest amongst the six pariahs[42] — whose value lies midway between primes (10939, 10949), the latter with a prime index of ,[43] which is the nineteenth tetrahedral number.[44]
  • On the other hand, the Tits group , as the only non-strict
    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

sporadic groups
.

Worth noting,

are added together, a sum of 19 is obtained.

Science

The James Webb Space Telescope features a design of 19 hexagons.

Religion

Islam

Baháʼí faith

In the

Arabic: واحد, romanizedwāhid, lit.'one'). The numerical value of this word in the Abjad numeral system
is 19.

  • The Baháʼí calendar is structured such that a year contains 19 months of 19 days each (along with the intercalary period of Ayyám-i-Há), as well as a 19-year cycle and a 361-year (19x19) supercycle.
  • The Báb and his disciples formed a group of 19.
  • There were 19
    Apostles of Bahá'u'lláh
    .

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
    meantone tuning, being close to 1/3 comma meantone. See 19 equal temperament
    .
  • Some organs use the 19th harmonic to approximate a minor third.

Literature

Games

A 19x19 Go board
  • 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.

Age 19

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.

In other fields

  • The
    19th Amendment to the United States Constitution
    gave American women the right to vote.
  • The Vietnam War spanned over 19 years, from November 1955 to April 1975.

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.
  13. ^ Sloane, N. J. A. (ed.). "Sequence A111441 (Numbers k such that the sum of the squares of the first k primes is divisible by k)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2022-06-02.
  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. ^ Sloane, N. J. A. (ed.). "Sequence A068174 (Define an increasing sequence as follows. Start with an initial term, the seed (which need not have the property of the sequence); subsequent terms are obtained by inserting/placing at least one digit in the previous term to obtain the smallest number with the given property. Here the property is be a prime.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2022-07-26.
  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. ^ "Sloane's A125602 : Centered triangular numbers that are prime". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2016-05-31.
  21. ^ "Sloane's A003215 : Hex (or centered hexagonal) numbers". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2016-05-31.
  22. ^ Sloane, N. J. A. (ed.). "Sequence A000217 (Triangular numbers)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2022-07-13.
  23. ^ 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.
  24. ^ 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.
  25. ^ 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.
  26. ^ Trigg, C. W. (February 1964). "A Unique Magic Hexagon". Recreational Mathematics Magazine. Retrieved 2022-07-14.
  27. S2CID 218544330
    .
  28. ^ 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.
  29. .
  30. ^ Sloane, N. J. A. (ed.). "Sequence A072359 (Primes p such that the p-1 digits of the decimal expansion of k/p (for k equal to 1,2,3,...,p-1) fit into the k-th row of a magic square grid of order p-1.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-09-04.
  31. ^ Sloane, N. J. A. (ed.). "Sequence A000040 (The prime numbers)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-09-06.
  32. ^ 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.
  33. S2CID 120672023
    .
  34. .
  35. .
  36. .
  37. ^ Sloane, N. J. A. (ed.). "Sequence A002267 (The 15 supersingular primes.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2022-12-11.
  38. .
  39. .
    List of standard generators of all sporadic groups.
  40. .
  41. .
  42. ^ Sloane, N. J. A. (ed.). "Sequence A000040 (The prime numbers.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2024-02-28.
  43. ^ Sloane, N. J. A. (ed.). "Sequence A000292". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2024-02-28.
  44. ^ 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.
  45. 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.
  46. . ...for all groups of Lie type, including the twisted groups of Steinberg, Suzuki and Ree (and the Tits group).
  47. ^ Brigid: Triple Goddess of the Flame (Health, Hearth, & Forge)
  48. ^ 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.

External links