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

'''73''' ('''seventy-three''') is the [[Natural number|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.

== In mathematics ==
'''73''' is the 21st [[Prime number|prime number]], and [[Emirp|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|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|2]], [[3|3]], [[4|4]], [[5|5]], [[6|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|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|repunit]] in [[Octal|octal]] (111<sub>8</sub>). It is 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>

=== Sheldon prime ===

Notably, 73 is the only ''Sheldon prime'' to contain both "mirror" and "product" properties:<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, as an emirp, has 37 as its [[Duality (mathematics)|dual]] [[Permutable prime|permutable prime]], a mirroring of its base ten digits, [[7|7]] and [[3|3]]. 73 is the 21st prime number, while 37 is the 12th, which is a second mirroring; and
*73 has a prime [[Sequence|index]] of 21 = 7 × 3; a product property where the product of its base-10 digits is precisely its index in the [[List of prime numbers#The first 1000 prime numbers|sequence of prime numbers]].

[[Arithmetic]]ally, from sums of 73 and 37 with their prime indexes, one obtains:
{{block indent | em = 1.5 | text = 73 + 21 = 94 (or, 47 × 2),}}
{{block indent | em = 1.5 | text = 37 + 12 = 49 (or, 47 + 2 = 7<sup>2</sup>);}}
{{block indent | em = 1.5 | text = 94 − 49 = 45 (or, 47 − 2).}} 

Meanwhile,

[[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|permutable prime]], is the preceding consecutive star number (up to <span style="color:green">green</span> dots).]]

{{Bullet list
| 73 and 37 are [[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> They are also 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 star numbers and equivalently consecutive centered dodecagonal ('''12'''-gonal) numbers, respectively the 4th and the 3rd.<ref name="star number" />
| 73 and 37 are successive 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 (see [[Waring's problem]]).<ref>{{Cite OEIS |A002804 |(Presumed) solution to Waring's problem }}</ref>
| 73 and 37 are consecutive primes in the 7-integer [[Covering set|covering set]] of the first known [[Sierpinski number]] 78,557, of the form <math>k \times 2^m + 1</math> [[Composite number|composite]] for all natural numbers <math>m</math>, with 73 as its largest group member: {3, 5, 7, 13, 19, 37, 73}.<br />
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|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: {1, ''6'', 9, '''12''', 15, 16, '''21''', 22, ''24'', and 27}, with 37 present alongside 73. In particular, <math>A(n)</math> ≥ 73 for any <math>n</math>.
{{pb}}
| 73 and 37 have a [[Interval (mathematics)|range]] of 37 numbers, inclusive of both 37 and 73; their [[Subtraction|difference]], on the other hand, is [[36 (number)|36]], or thrice '''12'''. Also,

{{math|1=[[777 (number)|777]] = [[3|3]] × [[37 (number)|37]] × [[7|7]] = 21 × 37}}, where 37 is a [[Concatenation|concatenation]] of [[3|3]] and [[7|7]].

{{math|1=703}} equals the sum of the first 37 non-zero positive integers, equivalently the 37th [[Triangular number|triangular number]].<ref>{{Cite OEIS |A000217 |Triangular numbers. |access-date=2022-10-13 }}</ref> The [[Harmonic mean|harmonic mean]] of its divisors is {{math|3.7}}.

{{math|1=373}} has a prime [[Sequence|index]] of [[74 (number)|74]], or ''twice'' 37.<ref>{{Cite OEIS |A000040 |The prime numbers. |access-date=2022-10-13 }}</ref> Like 73 and 37, 373 is a [[Permutable prime|permutable prime]] alongside 337 and 733, the second of three trios of three-digit permutable primes in [[Decimal|decimal]].<ref>{{Cite OEIS |A003459 |Absolute primes (or permutable primes): every permutation of the digits is a prime. |access-date=2022-10-13 }}</ref> 337 is also the eighth star number.<ref name="star number" />
:{{math|1=337 + 373 + 733 = [[1000 (number)#1400 to 1499|1443]]}}, the number of edges in the [[Graph operations#Binary operations|join]] of ''two'' [[Cycle graph|cycle graph]]s of order 37.<ref>{{Cite web |url=https://oeis.org/A005563 | title=Sloane's A005563 : a(n) = n*(n+2) = (n+1)^2 – 1. |website=The On-Line Encyclopedia of Integer Sequences |publisher=OEIS Foundation |access-date=2022-10-15 |quote=Number of edges in the join of two cycle graphs, both of order n, C_n * C_n. }}</ref>

{{math|1=343 = 7 × 7 × 7 = 7<sup>'''3'''</sup>}}: the [[Cube (algebra)|''cube'']] of 7, or 7 ''cubed'', wherein replacing two neighboring digits with their [[Digit sum|digit sum]]s {{math|3 + 4}} and {{math|4 + 3}} yields {{math|3'''7''' ''':''' '''7'''3}}.
:Also, the product of neighboring digits {{math|1=3 × 4}} is [[12 (number)|12]], like {{math|1=4 × 3}}, while the sum of its prime factors {{math|1=7 + 7 + 7}} is {{math|1=21}}.

{{math|1=[[307 (number)|307]]}} has a prime index of [[63 (number)|63]], or thrice '''21'''. {{math|3 × 3 × 7}}, equivalently {{math|3 × 7 × 3}} and {{math|7 × 3 × 3}}, are all [[Permutation|permutation]]s of the prime factorization of 21.

}}

In [[Binary number|binary]] 73 is represented as {{math|1001001}}, while 21 in binary is {{math|10101}}, with 7 and 3 represented as {{math|111}} and {{math|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 {{math|21}} represents the [[Ternary numeral system|ternary]] (base-3) equivalent of the [[Decimal|decimal]] numeral [[7|7]], that is to say: {{math|1=21<sub>3</sub> = 7<sub>10</sub>}}.

=== Other properties ===

[[File:Lah numbers.svg|210px|thumb|[[Lah number|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|truncatable prime]]s in [[Decimal|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 |title=Combinatorial Identities |publisher={{wipe|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|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 (number theory)|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. |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|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 polytope operations.

In [[Monstrous moonshine|moonshine theory]] of [[Sporadic group|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|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 |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|integer matrix]] [[Definite quadratic form|definite quadratic]] that represents all [[Prime number|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>

== In science ==
*The [[Physics:Atomic number|atomic number]] of [[Chemistry:Tantalum|tantalum]]

=== In astronomy ===

* [[Astronomy:Messier object|Messier object]] [[Astronomy:Messier 73|M73]], a magnitude 9.0 apparent [[Astronomy:Open cluster|open cluster]] in the [[Astronomy:Constellation|constellation]] [[Astronomy:Aquarius (constellation)|Aquarius]].
* The [[Astronomy:New General Catalogue|New General Catalogue]] object NGC 73, a [[Astronomy:Barred spiral galaxy|barred spiral galaxy]] in the constellation [[Astronomy:Cetus|Cetus]].
* The number of seconds it took for the Space Shuttle ''Challenger'' OV-099 shuttle to explode after launch.
* 73 is the number of rows in the 1,679-bit [[Astronomy:Arecibo message|Arecibo message]], sent to space in search for extraterrestrial intelligence.

=== 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.

== In other fields ==
'''73''' is also:
*The number of books in the [[Religion:Catholic Bible|Catholic Bible]].<ref>{{Cite web|url=http://www.catholicbible101.com/thebible73or66books.htm | title=Catholic Bible 101|website=Catholic Bible 101|access-date=16 September 2018}}</ref>
*Amateur radio operators and other [[Engineering:Morse code|morse code]] users commonly use the number 73 as a "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 [[Engineering:Morse code|Morse code]], 73 is an easily recognized palindrome: ( - - · · ·  &nbsp;  · · · - - ).
*''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''.
*The registry of the U.S. Navy's nuclear aircraft carrier {{USS|George Washington|CVN-73}}, named after U.S. President George Washington.
*''No. 73'' was the name of a 1980s children's television programme in the United Kingdom. It ran from 1982 to 1988 and starred Sandi Toksvig.
*Pizza 73 is a Canadian pizza chain.
*Game show ''Match Game '73'' in 1973.
*Fender Rhodes Stage 73 Piano.
*Sonnet 73 by William Shakespeare.
*The number of the French department Savoie.
*On a CB radio, 10-73 means "speed trap at..."

== 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.)

==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=[[Social:IMDb|IMDb]] |url=https://www.imdb.com/title/tt1632225/characters/nm1433588|access-date=13 March 2023}}</ref> Jim Parsons was born in the year 1973.<ref>{{cite web|title=Jim Parsons|website=[[Social:IMDb|IMDb]] |url=https://www.imdb.com/name/nm1433588/}}</ref>  He 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}}

[[Category:Integers]]

{{Sourceattribution|73 (number)}}