← 72 73 74 →
Cardinalseventy-three
Ordinal73rd
(seventy-third)
Factorizationprime
Prime21st
Divisors1, 73
Greek numeralΟΓ´
Roman numeralLXXIII
Binary10010012
Ternary22013
Senary2016
Octal1118
Duodecimal6112
Hexadecimal4916

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.

In mathematics

73 is the 21st prime number, and emirp with 37, the 12th prime number.[1] It is also the eighth twin prime, with 71. 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: , and the smallest prime congruent to 1 modulo 24, as well as the only prime repunit in octal (1118). It is the fourth star number.[2]

Sheldon prime

Notably, 73 is the only Sheldon prime to contain both "mirror" and "product" properties:[3]

Arithmetically, from sums of 73 and 37 with their prime indexes, one obtains:

73 + 21 = 94 (or, 47 × 2),
37 + 12 = 49 (or, 47 + 2 = 72);
94 − 49 = 45 (or, 47 − 2).

Meanwhile,

73 as a star number (up to blue dots). 37, its dual permutable prime, is the preceding consecutive star number (up to green dots).

In binary 73 is represented as 1001001, while 21 in binary is 10101, with 7 and 3 represented as 111 and 11 respectively; all which are palindromic. Of the seven binary digits representing 73, there are three 1s. 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.

Other properties

Lah numbers for and between 1 and 4. The sum of values with and is 73.

73 is one of the fifteen left-truncatable and right-truncatable primes in decimal, meaning it remains prime when the last "right" digit is successively removed and it remains prime when the last "left" digit is successively removed; and because it is a twin prime (with 71), it is the only two-digit twin prime that is both a left-truncatable and right-truncatable prime.

The row sum of Lah numbers of the form with and is equal to .[14] 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 into any number of lists, where a list means an ordered subset.[15]

73 requires 115 steps to return to 1 in the Collatz problem, and 37 requires 21: {37, 112, 56, 28, 14, 7, 22, 11, 34, 17, 52, 26, 13, 40, 20, 10, 5, 16, 8, 4, 2, 1}.[16] 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.[17]

There are 73 three-dimensional arithmetic crystal classes that are part of 230 crystallographic space group types.[18] 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 simplex group, 23 from the demihypercube group, and 31 from the hypercubic group, of which 15 equivalent solutions are shared between and from distinct polytope operations.

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.[19] contains a total of 194 conjugacy classes that involve 73 distinct orders (without including multiplicities over which letters run).[20]

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}.[21]

In science

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In astronomy

In chronology

In other fields

73 is also:

In sports

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Popular culture

The Big Bang Theory

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.".[24] Jim Parsons was born in the year 1973.[25] He often wears a t-shirt with the number 73 on it.[26]

See also

References

  1. ^ "Sloane's A006567 : Emirps". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2016-05-29.
  2. ^ a b c "Sloane's A003154 : Centered 12-gonal numbers. Also star numbers: 6*n*(n-1) + 1". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2016-05-29.
  3. ^ Pomerance, Carl; Spicer, Chris (February 2019). "Proof of the Sheldon conjecture" (PDF). American Mathematical Monthly. 126 (8): 688–698. doi:10.1080/00029890.2019.1626672. S2CID 204199415.
  4. ^ Sloane, N. J. A. (ed.). "Sequence A031157 (Numbers that are both lucky and prime.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2022-10-14.
  5. ^ Sloane, N. J. A. (ed.). "Sequence A023201 (Primes p such that p + 6 is also prime. (Lesser of a pair of sexy primes.))". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2022-10-14.
  6. ^ 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 2022-10-14.
  7. ^ Sloane, N. J. A. (ed.). "Sequence A005109 (Class 1- (or Pierpont) primes)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2022-12-19.
  8. ^ Sloane, N. J. A. (ed.). "Sequence A002804 ((Presumed) solution to Waring's problem)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
  9. ^ Sloane, N. J. A. (ed.). "Sequence A305473 (Let k be a Sierpiński or Riesel number divisible by 2*n – 1...)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2022-10-13.
  10. ^ Sloane, N. J. A. (ed.). "Sequence A000217 (Triangular numbers.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2022-10-13.
  11. ^ Sloane, N. J. A. (ed.). "Sequence A000040 (The prime numbers.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2022-10-13.
  12. ^ Sloane, N. J. A. (ed.). "Sequence A003459 (Absolute primes (or permutable primes): every permutation of the digits is a prime.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2022-10-13.
  13. ^ "Sloane's A005563 : a(n) = n*(n+2) = (n+1)^2 – 1". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2022-10-15. Number of edges in the join of two cycle graphs, both of order n, C_n * C_n.
  14. ^ Riordan, John (1968). Combinatorial Identities. John Wiley & Sons. p. 194. LCCN 67031375. MR 0231725. OCLC 681863847.
  15. ^ Sloane, N. J. A. (ed.). "Sequence A000262 (Number of "sets of lists": number of partitions of {1,...,n} into any number of lists, where a list means an ordered subset.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-09-02.
  16. ^ 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 2023-09-18.
  17. ^ Sloane, N. J. A. "3x+1 problem". The On-Line Encyclopedia of Integer Sequences. The OEIS Foundation. Retrieved 2023-09-18.
  18. ^ Sloane, N. J. A. (ed.). "Sequence A004027 (Number of arithmetic n-dimensional crystal classes.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2022-11-29.
  19. ^ Sloane, N. J. A. (ed.). "Sequence A002267 (The 15 supersingular primes)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2022-10-13.
  20. ^ He, Yang-Hui; McKay, John (2015). "Sporadic and Exceptional". p. 20. arXiv:1505.06742 [math.AG].
  21. ^ Sloane, N. J. A. (ed.). "Sequence A154363 (Numbers from Bhargava's prime-universality criterion theorem)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
  22. ^ "Catholic Bible 101". Catholic Bible 101. Retrieved 16 September 2018.
  23. ^ "Ham Radio History".
  24. ^ "The Big Bang Theory (TV Series) - The Alien Parasite Hypothesis (2010) - Jim Parsons: Sheldon Cooper". IMDb. Retrieved 13 March 2023.
  25. ^ "Jim Parsons". IMDb.
  26. ^ "The Alien Parasite Hypothesis". The Big Bang Theory. Season 4. Episode 10.