# 73 (number)

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

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: 104 + 1 = 10,001 = 73 × 137, and the smallest prime congruent to 1 modulo 24, as well as the only prime repunit in base 8 (1118). It is the fourth star number.[2]

73 as a star number (up to blue dots). 37, its dual permutable prime, is the preceding consecutive star number (up to green dots) within the sequence of star numbers.[2]

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

• 73, as an emirp, has 37 as its dual permutable prime, a mirroring of its base ten digits, 7 and 3. 73 is the 21st prime number, while 37 is the 12th, which is a second mirroring; and
• 73 has a prime index of 21 = 7 × 3; a product property where the product of its base-10 digits is precisely its index in the sequence of prime numbers.

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).

Other properties ligating 73 with 37 include:

• 73 and 37 are lucky primes and sexy primes, both twice over.[4][5][6] They are also successive Pierpont primes, respectively the 9th and 8th.[7]
• 73 and 37 are consecutive star numbers and equivalently consecutive centered dodecagonal (12-gonal) numbers, respectively the 4th and the 3rd.[2]
• 73 and 37 are successive values of ${\displaystyle g(k)}$ such that every positive integer can be written as the sum of 73 or fewer sixth powers, or 37 or fewer fifth powers (see Waring's problem).[8]
• 73 and 37 are consecutive primes in the 7-integer covering set of the first known Sierpinski number 78,557, of the form ${\displaystyle k\times 2^{m}+1}$ composite for all natural numbers ${\displaystyle m}$, with 73 as its largest group member: {3, 5, 7, 13, 19, 37, 73}.

Consider the following sequence ${\displaystyle A(n)}$:[9]

Let ${\displaystyle k}$ be a Sierpiński number or Riesel number divisible by ${\displaystyle 2n-1}$, and let ${\displaystyle p}$ be the largest number in a set of primes which cover every number of the form ${\displaystyle k\times 2^{m}+1}$ or of the form ${\displaystyle k\times 2^{m}-1}$, with ${\displaystyle m\geq 1}$;
${\displaystyle A(n)}$ equals ${\displaystyle p}$ if and only if there exists no number ${\displaystyle k}$ that has a covering set with largest prime greater than ${\displaystyle p}$.
Known such index values ${\displaystyle n}$ where ${\displaystyle p}$ is equal to 73 as the largest member of such covering sets are: {1, 6, 9, 12, 15, 16, 21, 22, 24, and 27}, with 37 present alongside 73. In particular, ${\displaystyle A(n)}$ ≥ 73 for any ${\displaystyle n}$.
• 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.

777 = 3 × 37 × 7 = 21 × 37, where 37 is a concatenation of 3 and 7.

703 equals the sum of the first 37 non-zero positive integers, equivalently the 37th triangular number.[10] The harmonic mean of its divisors is 3.7.

373 has a prime index of 74, or twice 37.[11] Like 73 and 37, 373 is a permutable prime alongside 337 and 733, the second of three trios of three-digit permutable primes in decimal.[12] 337 is also the eighth star number.[2]

337 + 373 + 733 = 1443, the number of edges in the join of two cycle graphs of order 37.[13]

343 = 7 × 7 × 7 = 73: the cube of 7, or 7 cubed, wherein replacing two neighboring digits with their digit sums 3 + 4 and 4 + 3 yields 37 : 73.

Also, the product of neighboring digits 3 × 4 is 12, like 4 × 3, while the sum of its prime factors 7 + 7 + 7 is 21.
307 has a prime index of 63, or thrice 21. 3 × 3 × 7, equivalently 3 × 7 × 3 and 7 × 3 × 3, are all permutations of the prime factorization of 21.

In binary, 73 is 1001001, while 21 in binary is 10101, and 7 in binary is 111; all which are palindromic. Of the 7 binary digits representing 73, there are 3 ones. 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.

There are 73 three-dimensional arithmetic crystal classes that are part of 230 crystallographic space group types.[14] 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 ${\displaystyle \mathrm {A} _{5}}$ simplex group, 23 from the ${\displaystyle \mathrm {D} _{5}}$ demihypercube group, and 31 from the ${\displaystyle \mathrm {B} _{5}}$ hypercubic group, of which 15 equivalent solutions are shared between ${\displaystyle \mathrm {D} _{5}}$ and ${\displaystyle \mathrm {B} _{5}}$ from distinct polytope operations.

In moonshine theory of sporadic groups, 73 is the first non-supersingular prime greater than 71. 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.[15]

73 is the largest member of the 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}.[16]

## In science

### In chronology

• The year AD 73, 73 BC, or 1973.
• The number of days in 1/5 of a non-leap year.
• The 73rd day of a non-leap year is March 14, also known as Pi Day.

73 is also:

## In sports

• In international curling competitions, each side is given 73 minutes to complete all of its throws.
• In baseball, the single-season home run record set by Barry Bonds in 2001.
• In basketball, the number of games the Golden State Warriors won in the 2015–16 season (73-9), the most wins in NBA history.
• NFL: In the 1940 NFL championship game, the Bears beat the Redskins 73–0, the largest score ever in an NFL game. (The Redskins won their previous regular season game, 7–3).

## References

1. ^ "Sloane's A006567 : Emirps". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2016-05-29.
2. ^ a b c d "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. ^ 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.
15. ^ Sloane, N. J. A. (ed.). "Sequence A002267 (The 15 supersingular primes)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2022-10-13.
16. ^
17. ^ "Catholic Bible 101". Catholic Bible 101. Retrieved 16 September 2018.
18. ^