19 (number)
[3] 19 is the fifth central trinomial coefficient,[4] and the maximum number of fourth powers needed to sum up to any natural number (see, Waring's problem).
[6] 19 is the eighth strictly non-palindromic number in any base, following 11 and preceding 47.
In the Engel expansion of pi,[9] 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.
[13] 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.
[19] The sum of the squares of the first nineteen primes is divisible by 19.
[21][22] 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.
[30] 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: 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.
[36] 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.
a polynomial with a total of twenty coefficients, which specifies a space for cubic surfaces that is 19-dimensional.
It is the middle indexed member in the sequence of fifteen such primes that divide the order of the Friendly Giant
[40] 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.
In the Bábí and Baháʼí Faiths, a group of 19 is called a Váhid, a Unity (Arabic: واحد, romanized: wāhid, lit. 'one').
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.
