# 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: ${\displaystyle 10^{4}+1=10,001=73\times 137}$, 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

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]

• 73 has 37 as the mirroring of its decimal digits. 73 is the 21st prime number, and 37 the 12th. The "mirror property" is fulfilled when 73 has a mirrored permutation of its digits (37) that remains prime. Similarly, their respective prime indices (21 and 12) in the list of prime numbers are also permutations of the same digits (1, and 2).
• 73 is the 21st prime number. It satisfies the "product property" since the product of its decimal digits is precisely in equivalence with its index in the sequence of prime numbers. i.e., 21 = 7 × 3. On the other hand, 37 does not fulfill the product property, since, naturally, its digits also multiply to 21; therefore, the only number to fulfill this property between these two numbers is 73, and as such it is the only "Sheldon prime".

### Further properties ligating 73 and 37

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 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,

• 777 = 3 × 37 × 7 = 21 × 37, where 37 is a concatenation of 3 and 7. 777 is a polite number, in equivalence with a sum of 37 consecutive integers, 3 + ... + 39.
• 703 equals the sum of the first 37 non-zero positive integers, equivalently the 37th triangular number.[4] The harmonic mean of its divisors is 3.7.
• 373 has a prime index of 74, or twice 37.[5] 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.[6] 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.[7]
• 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.

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).[2] They are successive lucky primes and sexy primes, both twice over,[9][10][11] and successive Pierpont primes, respectively the 9th and 8th.[12] 73 and 37 are consecutive 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 (and 19 or fewer fourth powers; see Waring's problem).[13]

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.

#### Sierpiński numbers

73 and 37 are consecutive primes in the seven-integer covering set of the first known Sierpiński number 78,557 of the form ${\displaystyle k\times 2^{n}+1}$ that is composite for all natural numbers ${\displaystyle n}$, where 73 is the largest member: ${\displaystyle \{3,5,7,13,19,37,73\}.}$ More specifically, ${\displaystyle 78,557\times 2^{n}+1}$ modulo 36 will be divisible by at least one of the integers in this set.

Consider the following sequence ${\displaystyle A(n)}$:[14]

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: ${\displaystyle \{1,6,9,12,15,16,21,22,24,27\}}$, with 37 present alongside 73. In particular, ${\displaystyle A(n)}$ ≥ 73 for any ${\displaystyle n}$.

In addition, 73 is the largest member in the covering set ${\displaystyle \{5,7,13,73\}}$ of the smallest proven generalized Sierpiński number of the form ${\displaystyle k\times b^{n}+1}$ in nonary ${\displaystyle (2,344\times 9^{n}+1)}$, while it is also the largest member of the covering set ${\displaystyle \{7,11,13,73\}}$ that belongs to the smallest such provable number in decimal ${\displaystyle (9,175\times 10^{n}+1)}$ — both in congruencies ${\displaystyle {\text{mod }}6}$.[15][16]

### Other properties

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 ${\displaystyle L(n,k)=\textstyle {\left\lfloor {n \atop k}\right\rfloor }}$ with ${\displaystyle n=4}$ and ${\displaystyle k={1,2,3,4}}$ is equal to ${\displaystyle 73}$.[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 ${\displaystyle \{{1,2,3,4}\}}$ into any number of lists, where a list means an ordered subset.[18]

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}.[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 ${\displaystyle \mathrm {A} _{5}}$ 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 that does not divide the order of the largest sporadic group ${\displaystyle \mathrm {F_{1}} }$. 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.[22] ${\displaystyle \mathrm {F_{1}} }$ 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.

## 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.)

### Doctor Who

In a 2024 episode of Doctor Who, "73 Yards", the character Ruby Sunday is haunted by a mysterious woman who is always standing exactly 73 yards away from her.

### 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.".[30] Jim Parsons was born in the year 1973.[31] He often wears a t-shirt with the number 73 on it.[32]

## 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 A000217 (Triangular numbers.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2022-10-13.
5. ^ Sloane, N. J. A. (ed.). "Sequence A000040 (The prime numbers.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2022-10-13.
6. ^ 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.
7. ^ "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.
8. ^ a b Sloane, N. J. A. (ed.). "Sequence A002061 (Central polygonal numbers: a(n) equal to n^2 - n + 1.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2024-02-29.
9. ^ 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.
10. ^ 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.
11. ^ 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.
12. ^ 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.
13. ^ Sloane, N. J. A. (ed.). "Sequence A002804 ((Presumed) solution to Waring's problem)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
14. ^ 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.
15. ^ Brunner, Amy; Caldwell, Chris K.; Krywaruczenko, Daniel; Lownsdale, Chris (2009). "Generalized Sierpiński Numbers to Base b" (PDF). 数理解析研究所講究録 [Notes from the Institute of Mathematical Analysis] (New Aspects of Analytic Number Theory). 1639. Kyoto: RIMS: 69–79. hdl:2433/140555. S2CID 38654417.
16. ^ Gary Barnes (December 2007). "Sierpinski conjectures and proofs (Conjectures 'R Us Project)". No Prime Left Behind (NPLB). Retrieved 2024-03-10.
17. ^ Riordan, John (1968). Combinatorial Identities. John Wiley & Sons. p. 194. LCCN 67031375. MR 0231725. OCLC 681863847.
18. ^ 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.
19. ^ 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.
20. ^ Sloane, N. J. A. "3x+1 problem". The On-Line Encyclopedia of Integer Sequences. The OEIS Foundation. Retrieved 2023-09-18.
21. ^ 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.
22. ^ Sloane, N. J. A. (ed.). "Sequence A002267 (The 15 supersingular primes)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2022-10-13.
23. ^ He, Yang-Hui; McKay, John (2015). "Sporadic and Exceptional". p. 20. arXiv:1505.06742 [math.AG].
24. ^
25. ^ "Tantalum". Royal Society of Chemistry. 4 May 2024. Retrieved 4 May 2024.
26. ^ admin (2015-04-30). "Messier 37". Messier Objects. Retrieved 2024-05-25.
27. ^ "Arecibo Message". SETI Institute. Retrieved 2024-05-25.
28. ^ "Catholic Bible 101". Catholic Bible 101. Retrieved 16 September 2018.
29. ^
30. ^ "The Big Bang Theory (TV Series) - The Alien Parasite Hypothesis (2010) - Jim Parsons: Sheldon Cooper". IMDb. Retrieved 13 March 2023.
31. ^
32. ^ "The Alien Parasite Hypothesis". The Big Bang Theory. Season 4. Episode 10.