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.
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,
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 twocycle 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 sums3 + 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]
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 that is
composite for all natural numbers , where 73 is the largest member: More specifically, modulo36 will be divisible by at least one of the integers in this set.
Let be a Sierpiński number or
Riesel number divisible by , and let be the largest number in a set of primes which cover every number of the form or of the form , with ;
equals if and only if there exists no number that has a covering set with largest prime greater than .
Known such index values where is equal to 73 as the largest member of such covering sets are: , with 37 present alongside 73. In particular, ≥ 73 for any .
In addition, 73 is the largest member in the covering set of the smallest proven generalized Sierpiński number of the form in
nonary, while it is also the largest member of the covering set that belongs to the smallest such provable number in
decimal — both in congruencies .[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 with and is equal to .[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 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
isomorphicpoint, with the remaining
157 groups nonsymmorphic (the 37th prime is 157).
73 is the largest member of a 17-
integer matrixdefinite 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.
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.[26] In
Morse code, 73 is an easily recognized palindrome: ( - - · · · · · · - - ).
73 (also known as 73 Amateur Radio Today) was an
amateur radio magazine published from 1960 to 2003.
73 was the number on the Torpedo Patrol (PT) boat in the TV show McHale's Navy.
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.)
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.".[27]Jim Parsons was born in the year
1973.[28] He often wears a
t-shirt with the number 73 on it.[29]
^"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.
^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.
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