Sheldon prime

Notably, 73 is the only Sheldon prime to contain both "mirror" and "product" properties:^[3]

• 73, as an emirp, has 37 as its dual permutable prime, a mirroring of its base ten digits, 7 and 3. 73 is the 21st prime number, while 37 is the 12th, which is a second mirroring; and
• 73 has a prime index of 21 = 7 × 3; a product property where the product of its base-10 digits is precisely its index in the sequence of prime numbers.

Arithmetically, from sums of 73 and 37 with their prime indexes, one obtains:

Meanwhile,

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

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.

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 ${\displaystyle 73}$.^[14] 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.^[15]

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}.^[16] 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.^[17]

There are 73 three-dimensional arithmetic crystal classes that are part of 230 crystallographic space group types.^[18] 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.^[19] ${\displaystyle \mathrm {F_{1}} }$ contains a total of 194 conjugacy classes that involve 73 distinct orders (without including multiplicities over which letters run).^[20]

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}.^[21]

In astronomy

• Messier object M73, a magnitude 9.0 apparent open cluster in the constellation Aquarius.
• The New General Catalogue object NGC 73, a barred spiral galaxy in the constellation 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 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.

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.".^[24] Jim Parsons was born in the year 1973.^[25] He often wears a t-shirt with the number 73 on it.^[26]