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{{Infobox number
| number = 73
| factorization = [[prime number|prime]]
| prime = 21st
| divisor = 1, 73
}}

'''73''' ('''seventy-three''') is the [[natural number]] following [[72 (number)|72]] and preceding [[74 (number)|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]].<ref name="star number">{{Cite web|url=https://oeis.org/A003154|title=Sloane's A003154 : Centered 12-gonal numbers. Also star numbers: 6*n*(n-1) + 1|website=The On-Line Encyclopedia of Integer Sequences|publisher=OEIS Foundation|access-date=2016-05-29}}</ref> It is also the eighth [[twin prime]], with [[71 (number)|71]].

== In mathematics ==
73 is the 21st [[prime number]], and [[emirp]] with [[37 (number)|37]], the 12th prime number.<ref>{{Cite web|url=https://oeis.org/A006567|title=Sloane's A006567 : Emirps|website=The On-Line Encyclopedia of Integer Sequences|publisher=OEIS Foundation|access-date=2016-05-29}}</ref> It is also the eighth [[twin prime]], with [[71 (number)|71]]. It is the largest minimal [[Primitive root modulo n|primitive root]] in the first 100,000 primes; in other words, if {{mvar|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 {{mvar|p}}. 73 is also the smallest factor of the first [[Composite number|composite]] [[generalized Fermat number]] in [[decimal]]: <math>10^{4}+1=10,001=73\times 137</math>, and the smallest prime [[Modular arithmetic#Congruence|congruent to]] 1 modulo [[24 (number)|24]], as well as the only prime [[repunit]] in [[octal]] (111<sub>8</sub>). It is the fourth [[star number]].<ref name="star number"></ref>

=== Sheldon prime ===
Where 73 and [[37 (number)|37]] are part of the sequence of [[permutable prime]]s and [[emirp]]s 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:<ref>{{Cite journal| last1=Pomerance|first1=Carl|last2=Spicer|first2=Chris|date=February 2019|title=Proof of the Sheldon conjecture|url=https://math.dartmouth.edu/~carlp/sheldon02132019.pdf|journal=American Mathematical Monthly|volume=126 |issue=8 |pages=688–698 |doi=10.1080/00029890.2019.1626672|s2cid=204199415 }}</ref>

* 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 [[List of prime numbers#The first 1000 prime numbers|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 ===

[[File:Star number 73 as sum of gnomons.svg|175px|right|thumb|'''73''' as a [[star number]] (up to <span style="color:blue">blue</span> dots). [[37 (number)|37]], its dual [[permutable prime]], is the preceding consecutive star number (up to <span style="color:green">green</span> dots).]]

73 and 37 are also consecutive [[star number]]s, equivalently consecutive [[centered dodecagonal number|centered dodecagonal]] ('''12'''-gonal) numbers (respectively the 4th and the 3rd).<ref name="star number" /> They are successive [[lucky number|lucky primes]] and [[sexy primes]], both twice over,<ref>{{Cite OEIS |A031157 |Numbers that are both lucky and prime. |access-date=2022-10-14 }}</ref><ref>{{Cite OEIS |A023201 |Primes p such that p + 6 is also prime. (Lesser of a pair of sexy primes.) |access-date=2022-10-14 }}</ref><ref>{{Cite OEIS |A046117 |Primes p such that p-6 is also prime. (Upper of a pair of sexy primes.) |access-date=2022-10-14 }}</ref> and successive [[Pierpont prime]]s, respectively the 9th and 8th.<ref>{{Cite OEIS |A005109 |Class 1- (or Pierpont) primes |access-date=2022-12-19 }}</ref> 73 and 37 are consecutive values of <math>g(k)</math> such that every [[positive integer]] can be written as the [[summation|sum]] of 73 or fewer sixth powers, or 37 or fewer fifth powers (and 19 or fewer [[fourth power]]s; see [[Waring's problem]]).<ref>{{Cite OEIS |A002804 |(Presumed) solution to Waring's problem }}</ref>

73 and 37 are consecutive primes in the seven-integer [[covering set]] of the first known [[Sierpiński number]] 78,557 of the form <math>k \times 2^{n} + 1</math> that is [[composite number|composite]] for all natural numbers <math>n</math>, where 73 is the largest member: <math>\{3, 5, 7, 13, 19, 37, 73\}.</math> More specifically, <math>78,557 \times 2^{n} + 1</math> [[Modular arithmetic|modulo]] [[36 (number)|36]] will be divisible by at least one of the integers in this set.{{cn|date=February 2025}}

Consider the following sequence <math>A(n)</math>:<ref>{{Cite OEIS |A305473 |Let k be a Sierpiński or Riesel number divisible by 2*n – 1... |access-date=2022-10-13 }}</ref>
:Let <math>k</math> be a Sierpiński number or [[Riesel number]] divisible by <math>2n - 1</math>, and let <math>p</math> be the largest number in a set of primes which cover every number of the form <math>k \times 2^{m} + 1</math> or of the form <math>k \times 2^{m} - 1</math>, with <math>m \geq 1</math>;
:<math>A(n)</math> equals <math>p</math> [[if and only if]] there exists no number <math>k</math> that has a covering set with largest prime greater than <math>p</math>.
Known such index values <math>n</math> where <math>p</math> is equal to 73 as the largest member of such covering sets are: <math>\{1, 6, 9, 12, 15, 16, 21, 22, 24, 27\}</math>, with 37 present alongside 73. In particular, <math>A(n)</math> ≥ 73 for any <math>n</math>.
{{pb}}

In addition, 73 is the largest member in the covering set <math>\{5,7,13,73\}</math> of the smallest proven '''generalized Sierpiński number''' of the form <math>k \times b^{n} + 1</math> in [[nonary]] <math>(2,344 \times 9^{n} + 1)</math>, while it is also the largest member of the covering set <math>\{7, 11, 13, 73\}</math> that belongs to the smallest such provable number in [[decimal]] <math>(9,175 \times 10^n + 1)</math> — both in congruencies <math>\text {mod } 6</math>.<ref>{{Cite journal |url=https://www.kurims.kyoto-u.ac.jp/~kyodo/kokyuroku/contents/pdf/1639-08.pdf |title=Generalized Sierpiński Numbers to Base b|last1=Brunner|first1=Amy|last2= Caldwell|first2= Chris K.|last3= Krywaruczenko|first3= Daniel|last4= Lownsdale|first4= Chris |journal=数理解析研究所講究録 [Notes from the Institute of Mathematical Analysis] (New Aspects of Analytic Number Theory) |publisher=[[Research Institute for Mathematical Sciences|RIMS]] |location=Kyoto |volume=1639 |year=2009 |pages=69–79 |hdl=2433/140555 |s2cid=38654417 }}</ref><ref>{{Cite web |author=Gary Barnes |title=Sierpinski conjectures and proofs (Conjectures 'R Us Project) |url=http://www.noprimeleftbehind.net/crus/Sierp-conjectures.htm |website=No Prime Left Behind (NPLB) |date=December 2007 |access-date=2024-03-10 }}</ref>

=== Other properties ===
[[File:Lah numbers.svg|210px|thumb|[[Lah number]]s for <math>n</math> and <math>k</math> between 1 and 4. The sum of values with <math>n = 4</math> and <math>k = \{1,2,3,4\}</math> is '''73'''. ]]

73 is one of the fifteen left-truncatable and right-[[truncatable prime]]s 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 number]]s of the form <math>L(n,k) = \textstyle {\left\lfloor {n \atop k} \right\rfloor}</math> with <math>n = 4</math> and <math>k = {1, 2, 3, 4}</math> is equal to <math>73</math>.<ref>{{Cite book |last=Riordan |first=John |author-link=John Riordan (mathematician) |title=Combinatorial Identities |publisher=[[John Wiley & Sons]] |year=1968 |page=194 |oclc=681863847 |lccn=67031375 |mr=0231725 |url=https://archive.org/details/combinatorialide00john|url-access=registration }}</ref> These numbers represent [[coefficient]]s expressing [[Falling and rising factorials|rising factorial]]s in terms of falling factorials, and vice-versa; equivalently in this case to the number of [[Partition of a set|partitions]] of <math>\{{1,2,3,4}\}</math> into any number of lists, where a list means an [[Order theory#Subsets of ordered sets|ordered subset]].<ref>{{Cite OEIS |A000262 |Number of "sets of lists": number of partitions of {1,...,n} into any number of lists, where a list means an ordered subset. |access-date=2023-09-02 }}</ref>

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'''}.<ref>{{Cite OEIS |A006577 |Number of halving and tripling steps to reach 1 in '3x+1' problem, or -1 if 1 is never reached. |access-date=2023-09-18 }}</ref> Collectively, the sum between these steps is [[136 (number)|136]], the 16th triangular number, where {16, 8, 4, 2, 1} is the only possible step root pathway.<ref>{{Cite web |last=Sloane |first= N. J. A. |author-link=Neil Sloane |url=http://oeis.org/wiki/3x%2B1_problem |title=3x+1 problem |website=The [[On-Line Encyclopedia of Integer Sequences]] |publisher=The OEIS Foundation |access-date=2023-09-18 }}</ref>

There are 73 three-dimensional [[Space group#In other dimensions|arithmetic crystal classes]] that are part of 230 crystallographic space group types.<ref>{{Cite OEIS |A004027 |Number of arithmetic n-dimensional crystal classes. |access-date=2022-11-29 }}</ref> These 73 groups are specifically '''symmorphic''' groups such that all operating lattice symmetries have one common fixed [[Isomorphism|isomorphic]] [[Point group|point]], with the remaining [[157 (number)|157]] groups nonsymmorphic (the 37th prime is 157). 

In [[five-dimensional space]], there are 73 [[Euclidean space|Euclidean solutions]] of [[Uniform 5-polytope|5-polytopes]] with [[Uniform 5-polytope|uniform]] symmetry, excluding [[Uniform 5-polytope#Enumerating the convex uniform 5-polytopes|prismatic forms]]: [[19 (number)|19]] from the <math>\mathrm A_{5}</math> [[A5 polytope|simplex group]], [[23 (number)|23]] from the <math>\mathrm D_{5}</math> [[D5 polytope|demihypercube group]], and [[31 (number)|31]] from the <math>\mathrm B_{5}</math> [[B5 polytope|hypercubic group]], of which 15 equivalent solutions are shared between <math>\mathrm D_{5}</math> and <math>\mathrm B_{5}</math> from distinct [[Uniform polytope#Operations|polytope operations]].

In [[Monstrous moonshine|moonshine theory]] of [[sporadic group]]s, 73 is the first [[Supersingular prime (moonshine theory)|''non-supersingular prime'']] greater than 71 that does not divide the [[Order (mathematics)|order]] of the largest sporadic group <math>\mathrm {F_{1}}</math>. 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.<ref>{{Cite OEIS |A002267 |The 15 supersingular primes |access-date =2022-10-13 }}</ref> <math>\mathrm {F_{1}}</math> contains a total of 194 [[conjugacy class]]es that involve 73 distinct orders (without including [[Multiplicity (mathematics)|multiplicities]] over which letters run).<ref>{{Cite arXiv |last1=He |first1=Yang-Hui |last2=McKay |first2=John |author1-link=Yang-Hui He |author2-link=John McKay (mathematician) |title=Sporadic and Exceptional |year=2015 |pages=20 |class=math.AG |eprint=1505.06742 }}</ref>

73 is the largest member of a 17-[[integer matrix]] [[Definite quadratic form|definite quadratic]] that represents all [[prime number]]s: {2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, '''37''', 41, 43, 47, 67, '''73'''},<ref>{{Cite OEIS |A154363 |Numbers from Bhargava's prime-universality criterion theorem }}</ref> with consecutive primes between [[2]] through [[47 (number)|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.<ref name="CentrPN">{{Cite OEIS |A002061 |Central polygonal numbers: a(n) equal to n^2 - n + 1. |access-date=2024-02-29 }}</ref>

== In other fields ==
'''73''' is also:
*[[Amateur radio operator]]s and other [[morse code]] users commonly use the number 73 as a [[92 Code|"92 Code" abbreviation]] for "best regards", typically when ending a [[Q code|QSO]] (a conversation with another operator). These codes also facilitate communication between operators who may not be native English speakers.<ref>{{cite web | url=http://www.arrl.org/ham-radio-history | title=Ham Radio History }}</ref> In [[Morse code]], 73 is an easily recognized palindrome: ( - - · · ·  &nbsp;  · · · - - ).
*On a [[Citizens band radio|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''.<ref>{{cite web|title=The Big Bang Theory (TV Series) - The Alien Parasite Hypothesis (2010) - Jim Parsons: Sheldon Cooper|website=[[IMDb]] |url=https://www.imdb.com/title/tt1632225/characters/nm1433588|access-date=13 March 2023}}</ref> [[Jim Parsons]], who plays Cooper in the series, was born in [[1973]].<ref>{{cite web|title=Jim Parsons|website=[[IMDb]] |url=https://www.imdb.com/name/nm1433588/}}</ref>  His character often wears a [[t-shirt]] with the number 73 on it.<ref>{{cite episode|title=The Alien Parasite Hypothesis|series=The Big Bang Theory|season=4|number=10}}</ref>

== See also ==
* [[List of highways numbered 73]]

== References ==
{{Reflist}}
{{Integers|zero}}

[[Category:Integers]]