73 (number)

← 72 73 74 →
← 70 71 72 73 74 75 76 77 78 79 → List of numbers Integers ← 0 10 20 30 40 50 60 70 80 90 →
Cardinal	seventy-three
Ordinal	73rd (seventy-third)
Factorization	prime
Prime	21st
Divisors	1, 73
Greek numeral	ΟΓ´
Roman numeral	LXXIII, lxxiii
Binary	10010012
Ternary	22013
Senary	2016
Octal	1118
Duodecimal	6112
Hexadecimal	4916

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 21st prime number and the fourth star number. ^[1] It is also the eighth twin prime, with 71.

In mathematics

73 is the 21st prime number, and emirp with 37, the 12th prime number. ^[2] 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. ^[1]

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 TV character 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".

Other connections between 37 and 73

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

73 and 37 are also consecutive star numbers, equivalently consecutive centered dodecagonal (12-gonal) numbers (respectively the 4th and the 3rd). ^[1] They are successive lucky primes and sexy primes, both twice over, ^{[4] [5] [6]} and successive Pierpont primes, respectively the 9th and 8th. ^[7] 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). ^[8]

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.^{[ citation needed ]}

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: ${\displaystyle \{1,6,9,12,15,16,21,22,24,27\}}$, 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 ${\displaystyle {\text{mod }}6}$. ^{[10] [11]}

Other properties

Lah numbers for ${\displaystyle n}$ and ${\displaystyle k}$ between 1 and 4. The sum of values with ${\displaystyle n=4}$ and ${\displaystyle k=\{1,2,3,4\}}$ 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 ${\displaystyle L(n,k)=\textstyle {\left\lfloor {n \atop k}\right\rfloor }}$ with ${\displaystyle n=4}$ and ${\displaystyle k={1,2,3,4}}$ is equal to ${\displaystyle 73}$. ^[12] 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. ^[13]

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}. ^[14] 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. ^[15]

There are 73 three-dimensional arithmetic crystal classes that are part of 230 crystallographic space group types. ^[16] 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. ^[17] ${\displaystyle \mathrm {F_{1}} }$ contains a total of 194 conjugacy classes that involve 73 distinct orders (without including multiplicities over which letters run). ^[18]

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}, ^[19] with consecutive primes between 2 through 47.

73 is the ninth member of the sequence of Hogben's central polygonal numbers, which enumerates the maximal number of interior regions formed by nine intersecting circles. ^[20]

In other fields

73 is also:

• Amateur radio operators and other morse code users commonly use the number 73 as a "92 Code" abbreviation for "best regards", typically when ending a QSO (a conversation with another operator). These codes also facilitate communication between operators who may not be native English speakers. ^[21] In Morse code, 73 is an easily recognized palindrome: ( - - · · ·   · · · - - ).
• On a CB radio, 10-73 means "speed trap at..."

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. ^[22] Jim Parsons, who plays Cooper in the series, was born in 1973. ^[23] His character often wears a t-shirt with the number 73 on it. ^[24]

See also

• List of highways numbered 73

References

1. "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.
2. "Sloane's A006567 : Emirps". 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.). "SequenceA031157(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.). "SequenceA023201(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.). "SequenceA046117(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.). "SequenceA005109(Class 1- (or Pierpont) primes)". The On-Line Encyclopedia of Integer Sequences . OEIS Foundation. Retrieved 2022-12-19.
8. Sloane, N. J. A. (ed.). "SequenceA002804((Presumed) solution to Waring's problem)". The On-Line Encyclopedia of Integer Sequences . OEIS Foundation.
9. Sloane, N. J. A. (ed.). "SequenceA305473(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. 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.
11. Gary Barnes (December 2007). "Sierpinski conjectures and proofs (Conjectures 'R Us Project)". No Prime Left Behind (NPLB). Retrieved 2024-03-10.
12. Riordan, John (1968). Combinatorial Identities . John Wiley & Sons. p. 194. LCCN   67031375. MR   0231725. OCLC   681863847.
13. Sloane, N. J. A. (ed.). "SequenceA000262(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.
14. Sloane, N. J. A. (ed.). "SequenceA006577(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.
15. Sloane, N. J. A. "3x+1 problem". The On-Line Encyclopedia of Integer Sequences . The OEIS Foundation. Retrieved 2023-09-18.
16. Sloane, N. J. A. (ed.). "SequenceA004027(Number of arithmetic n-dimensional crystal classes.)". The On-Line Encyclopedia of Integer Sequences . OEIS Foundation. Retrieved 2022-11-29.
17. Sloane, N. J. A. (ed.). "SequenceA002267(The 15 supersingular primes)". The On-Line Encyclopedia of Integer Sequences . OEIS Foundation. Retrieved 2022-10-13.
18. He, Yang-Hui; McKay, John (2015). "Sporadic and Exceptional". p. 20. arXiv: 1505.06742 [math.AG].
19. Sloane, N. J. A. (ed.). "SequenceA154363(Numbers from Bhargava's prime-universality criterion theorem)". The On-Line Encyclopedia of Integer Sequences . OEIS Foundation.
20. Sloane, N. J. A. (ed.). "SequenceA002061(Central polygonal numbers: a(n) equal to n^2 - n + 1.)". The On-Line Encyclopedia of Integer Sequences . OEIS Foundation. Retrieved 2024-02-29.
21. "Ham Radio History".
22. "The Big Bang Theory (TV Series) - The Alien Parasite Hypothesis (2010) - Jim Parsons: Sheldon Cooper". IMDb . Retrieved 13 March 2023.
23. "Jim Parsons". IMDb .
24. "The Alien Parasite Hypothesis". The Big Bang Theory. Season 4. Episode 10.