73 (number)

73 (seventy-three) is the natural number following 72 and preceding 74.

In English, it is the smallest natural number with twelve letters in its spelled out name.

It is the largest minimal primitive root in the first 100,000 primes; in other words, if p is one of the first one hundred thousand primes, then at least one of the numbers 2, 3, 4, 5, 6, ..., 73 is a primitive root modulo p. 73 is also the smallest factor of the first composite generalized Fermat number in decimal:

[1] Where 73 and 37 are part of the sequence of permutable primes and emirps in base-ten, the number 73 is more specifically the unique Sheldon prime, named as an homage to Sheldon Cooper and defined as satisfying "mirror" and "product" properties, where:[3] Arithmetically, from sums of 73 and 37 with their prime indexes, one obtains: Meanwhile, 73 and 37 have a range of 37 numbers, inclusive of both 37 and 73; their difference, on the other hand, is 36, or thrice 12.

Also, Where 73 is the ninth member of Hogben's central polygonal numbers, which enumerates the maximal number of interior regions formed by nine intersecting circles,[8] members in this sequence also include 307, 343, and 703 as the 18th, 19th, and 27th indexed numbers, respectively (where 18 + 19 = 37); while 3, 7 and 21 are also in this sequence, as the 2nd, 3rd, and 5th members.

[8] 73 and 37 are also consecutive star numbers, equivalently consecutive centered dodecagonal (12-gonal) numbers (respectively the 4th and the 3rd).

such that every positive integer can be written as the sum of 73 or fewer sixth powers, or 37 or fewer fifth powers (and 19 or fewer fourth powers; see Waring's problem).

In addition to having prime factors 7 and 3, the number 21 represents the ternary (base-3) equivalent of the decimal numeral 7, that is to say: 213 = 710.

73 and 37 are consecutive primes in the seven-integer covering set of the first known Sierpiński number 78,557 of the form

modulo 36 will be divisible by at least one of the integers in this set.

is equal to 73 as the largest member of such covering sets are:

In addition, 73 is the largest member in the covering set

of the smallest proven generalized Sierpiński number of the form

, while it is also the largest member of the covering set

that belongs to the smallest such provable number in decimal

The row sum of Lah numbers of the form

[17] These numbers represent coefficients expressing rising factorials in terms of falling factorials, and vice-versa; equivalently in this case to the number of partitions of

[19] Collectively, the sum between these steps is 136, the 16th triangular number, where {16, 8, 4, 2, 1} is the only possible step root pathway.

[20] There are 73 three-dimensional arithmetic crystal classes that are part of 230 crystallographic space group types.

[21] These 73 groups are specifically symmorphic groups such that all operating lattice symmetries have one common fixed isomorphic point, with the remaining 157 groups nonsymmorphic (the 37th prime is 157).

In five-dimensional space, there are 73 Euclidean solutions of 5-polytopes with uniform symmetry, excluding prismatic forms: 19 from the

hypercubic group, of which 15 equivalent solutions are shared between

In moonshine theory of sporadic groups, 73 is the first non-supersingular prime greater than 71 that does not divide the order of the largest sporadic group

All primes greater than or equal to 73 are non-supersingular, while 37, on the other hand, is the smallest prime number that is not supersingular.

contains a total of 194 conjugacy classes that involve 73 distinct orders (without including multiplicities over which letters run).

[23] 73 is the largest member of a 17-integer matrix definite quadratic that represents all prime numbers: {2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 67, 73},[24] with consecutive primes between 2 through 47.

73 is also: 73 is Sheldon Cooper's favorite number in The Big Bang Theory.

He first expresses his love for it in "The Alien Parasite Hypothesis", the 73rd episode of The Big Bang Theory.

[26] Jim Parsons, who plays Cooper in the series, was born in 1973.

[27] His character often wears a t-shirt with the number 73 on it.