73 (number)

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← 72 73 74 →
Cardinalseventy-three
Ordinal73rd
(seventy-third)
Factorizationprime
Prime21st
Divisors1, 73
Greek numeralΟΓ´
Roman numeralLXXIII
Binary10010012
Ternary22013
Senary2016
Octal1118
Duodecimal6112
Hexadecimal4916

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.

In mathematics[edit]

73 is the 21st prime number, and forms the eighth twin prime with 71. 73 is an emirp and permutable prime with 37,[1] the 12 prime number. Both 73 and 37 identify as lucky primes, and sexy primes twice over.

  • 73 is the fourth star number.[2]
  • 73 is the smallest prime congruent to 1 modulo 24.
  • 73 is the only prime repunit in base 8 (1118).
  • 73 is smallest factor of the first composite generalized Fermat number in base 10: 104 + 1 = 10,001 = 73 × 137.
  • 73 is largest minimal primitive root in the first 100,000 primes. In other words, if p is one of the first 100,000 primes, then at least one of the numbers 2, 3, 4, 5, 6, ..., 73 is a primitive root modulo p.
  • 73 is 1001001 in binary, while 21 in binary is 10101, and 7 in binary is 111; all which are palindromic. Of the 7 binary digits representing 73, there are 3 ones. 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.
73 as a star number (up to blue dots). 37, its dual permutable prime, is the preceding consecutive star number (up to green dots) within the sequence of star numbers.[2]

Most notably, 73 is the sole 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, a second mirroring. 73 has a prime index of 21 = 7 x 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:

73 + 21 = 94 (or, 47 × 2),
37 + 12 = 49 (or, 47 + 2 = 72);
94 - 49 = 45 (or, 47 - 2).

Other properties ligating 73 with 37 include:

  • 73 and 37 are consecutive star numbers, respectively the 4th and the 3rd.[2]
  • 73 and 37 are consecutive primes in the 7-integer covering set of the first known Sierpinski number 78,557, with 73 as its largest group member: {3, 5, 7, 13, 19, 37, 73}.
  • 73 and 37 are consecutive values of such that every positive integer can be written as the sum of 73 or fewer sixth powers, or 37 or fewer fifth powers (see Waring's problem).[4]
  • 73 and 37 belong to the positive definite quadratic 17-integer matrix representing all primes, where 73 is the largest member of the set: {2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 67, 73}.[5]
  • 73 and 37 have a range of 37 numbers; in other words inclusive of 37 and 73, they comprise an interval totaling 37 numbers.
777 = 3 x 37 x 7, where 37 is a concatenation of 3 and 7.
373 has a prime index of 74, or twice 37.
  • 343 = 73 (the cube of 7, or equivalently, 7 cubed), where replacing two neighboring digits with their sum yields 37 and 73.

73 is the first non-supersingular prime greater than 71. All primes greater than or equal to 73 are non-supersingular; while 37, on the other hand, is the smallest non-supersingular prime.

In science[edit]

In astronomy[edit]

In chronology[edit]

  • 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[edit]

73 is also:

In sports[edit]

  • 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).

See also[edit]

References[edit]

  1. ^ "Sloane's A006567 : Emirps". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2016-05-29.
  2. ^ a b c "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.
  3. ^ Pomerance, Carl; Spicer, Chris (April 2019). "Proof of the Sheldon conjecture" (PDF). Amer. Math. Monthly. to appear.
  4. ^ Sloane, N. J. A. (ed.). "Sequence A002804 ((Presumed) solution to Waring's problem)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
  5. ^ Sloane, N. J. A. (ed.). "Sequence A154363 (Numbers from Bhargava's prime-universality criterion theorem)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
  6. ^ "Catholic Bible 101". Catholic Bible 101. Retrieved 16 September 2018.