19 (number)

Prime properties

19 is the seventh Mersenne prime exponent. ^[10] It is the second Keith number, and more specifically the first Keith prime. ^[11] In decimal, 19 is the third full reptend prime, ^[12] 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). ^[13]

• 19 × 91 = 1729, the first Hardy-Ramanujan number or taxicab number, also a Harshad number in base-ten, as it's divisible by the sum of its digits, 19. ^{[14] [15]}

1729 is also the nineteenth dodecagonal number. ^[16]

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. ^[17] 100019 is the next such smallest prime number, by the insertion of a 1.

• Numbers of the form 10_n9 equivalent to 10^x + 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}. ^[18]

Otherwise, ${\displaystyle R_{19}}$ is the second base-10 repunit prime, short for the number ${\displaystyle 1111111111111111111}$. ^[19]

The sum of the squares of the first nineteen primes is divisible by 19. ^[20]

Figurate numbers and magic figures

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

• The 19th triangular number is 190, equivalently the sum of the first 19 non-zero integers, that is also the sixth centered nonagonal number. ^{[23] [13]}

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 19_n, 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]

Collatz problem

The Collatz sequence for nine requires nineteen steps to return to one, more than any other number below it. ^[34] 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}. ^[35]

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}. ^[40]

• 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. ^[41]

${\displaystyle \mathrm {J_{1}} }$ 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. ^{[42] [43]}

• ${\displaystyle 10944}$ 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 ^[44] — whose value lies midway between primes (10939, 10949), the latter with a prime index of ${\displaystyle 1330}$, ^[45] which is the nineteenth tetrahedral number. ^[46]
• 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. ^[47]

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. ^[48] If the Tits group is indeed included as a group of Lie type, ^[49] 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.