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

Mathematics

Nineteen is the eighth prime number.

Number theory

19 forms a twin prime with 17, a cousin prime with 23, and a sexy prime with 13. 19 is the fifth central trinomial coefficient, and the maximum number of fourth powers needed to sum up to any natural number (see, Waring's problem). It is the number of compositions of 8 into distinct parts. 19 is the eighth strictly non-palindromic number in any base, following 11 and preceding 47. 19 is also the second octahedral number, after 6, and the sixth Heegner number. In the Engel expansion of pi, 19 is the seventh term following {1, 1, 1, 8, 8, 17} and preceding {300, 1991, ...} . The sum of the first terms preceding 17 is in equivalence with 19, where its prime index (8) are the two previous members in the sequence.

Prime properties

19 is the seventh Mersenne prime exponent. It is the second Keith number, and more specifically the first Keith prime. In decimal, 19 is the third full reptend prime, and the first prime number that is not a permutable prime, as its reverse (91) is composite (where 91 is also the fourth centered nonagonal number). 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. 100019 is the next such smallest prime number, by the insertion of a 1. 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} . Otherwise, R_{19} is the second base-10 repunit prime, short for the number. The sum of the squares of the first nineteen primes is divisible by 19.

Figurate numbers and magic figures

19 is the third centered triangular number as well as the third centered hexagonal number. n = {1, 2, 3, 5, 7, 26, 27, 53, 147, 236, 248, 386, 401} . The number of nodes in regular hexagon with all diagonals drawn is nineteen. 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.

Collatz problem

The Collatz sequence for 9|nine requires nineteen steps to return to 1|one, more than any other number below it. On the other hand, nineteen requires twenty steps, like eighteen. Less than ten thousand, only thirty-one other numbers require nineteen steps to return to one: {56, 58, 60, 61, 352, 360, 362, 368, 369, 372, 373, 401, 402, 403, 2176, ..., and 2421} .

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. 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 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. n + 2 mirrors]] exist up through the seventeenth dimension, where there exists a unique figure with nineteen facets. It is expressed with the simple Dynkin diagram,. 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.

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, the largest sporadic group: {2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 41, 47, 59, 71}. (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 . 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. 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. If the Tits group is indeed included as a group of Lie type, 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.

Heegner number

19 is the sixth Heegner number. 67 and 163, respectively the 19th and 38th prime numbers, are the two largest Heegner numbers, of 9|nine total. The sum of the first six Heegner numbers 1, 2, 3, 7, 11, and 19 sum to the seventh member and fourteenth prime number, 43. All of these numbers are prime, aside from the 1|unit. In particular, 163 is relevant in moonshine theory.

Science

Religion

Islam

Baháʼí Faith

In the Bábí and Baháʼí Faiths, a group of 19 is called a Váhid, a Unity. 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.

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Age 19

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