21 (number)

From HandWiki
Short description: Natural number
← 20 21 22 →
Factorization3 × 7
Divisors1, 3, 7, 21
Greek numeralΚΑ´
Roman numeralXXI
Base 36L36

21 (twenty-one) is the natural number following 20 and preceding 22.

The current century is the 21st century AD, under the Gregorian calendar.

In mathematics

21 is:

  • a composite number, its proper divisors being 1, 3 and 7, and a deficient number as the sum of these divisors is less than the number itself.
  • a Fibonacci number as it is the sum of the preceding terms in the sequence, 8 and 13.[1]
  • the fifth Motzkin number.[2]
  • a triangular number,[3] because it is the sum of the first six natural numbers (1 + 2 + 3 + 4 + 5 + 6 = 21).
  • an octagonal number.[4]
  • a Padovan number, preceded by the terms 9, 12, 16 (it is the sum of the first two of these) in the padovan sequence.[5]
  • a Blum integer, since it is a semiprime with both its prime factors being Gaussian primes.[6]
  • the sum of the divisors of the first 5 positive integers (i.e., 1 + (1 + 2) + (1 + 3) + (1 + 2 + 4) + (1 + 5))
  • the smallest non-trivial example of a Fibonacci number whose digits are Fibonacci numbers and whose digit sum is also a Fibonacci number.
  • a Harshad number.[7]* a repdigit in base 4 (1114).
  • the smallest natural number that is not close to a power of 2, 2n, where the range of closeness is ±n.
  • the smallest number of differently sized squares needed to square the square.[8]
  • the largest n with this property: for any positive integers a,b such that a + b = n, at least one of [math]\displaystyle{ \tfrac{a}{b} }[/math] and [math]\displaystyle{ \tfrac{b}{a} }[/math] is a terminating decimal. See a brief proof below.

Note that a necessary condition for n is that for any a coprime to n, a and n - a must satisfy the condition above, therefore at least one of a and n - a must only have factor 2 and 5.

Let [math]\displaystyle{ A(n) }[/math] denote the quantity of the numbers smaller than n that only have factor 2 and 5 and that are coprime to n, we instantly have [math]\displaystyle{ \frac{\varphi(n)}{2} \lt A(n) }[/math].

We can easily see that for sufficiently large n, [math]\displaystyle{ A(n) \sim \frac{\log_2(n) \log_5(n)}{2} = \frac{\ln^{2}(n)}{2 \ln(2) \ln(5)} }[/math], but [math]\displaystyle{ \varphi(n) \sim \frac {n} {e^\gamma\; \ln \ln n} }[/math], [math]\displaystyle{ A(n) = o(\varphi(n)) }[/math] as n goes to infinity, thus [math]\displaystyle{ \frac{\varphi(n)}{2} \lt A(n) }[/math] fails to hold for sufficiently large n.

In fact, For every n > 2, we have

[math]\displaystyle{ A\lt 1 + \log_2(n) + \frac{3 \log_5(n)}{2} + \frac{\log_2(n) \log_5(n)}{2} }[/math]


[math]\displaystyle{ \varphi(n) \gt \frac {n} {e^\gamma\; \log \log n + \frac {3} {\log \log n}} }[/math]

so [math]\displaystyle{ \frac{\varphi(n)}{2} \lt }[/math] fails to hold when n > 273 (actually, when n > 33).

Just check a few numbers to see that '= 2, 3, 4, 5, 6, 7, 8, 9, 11, 12, 15, 21.

In science

Age 21

  • In thirteen countries, 21 is the age of majority. See also: Coming of age.
  • In eight countries, 21 is the minimum age to purchase tobacco products.
  • In seventeen countries, 21 is the drinking age.
  • In nine countries, it is the voting age.
  • In the United States:
    • 21 is the minimum age at which a person may gamble or enter casinos in most states (since alcohol is usually provided).
    • 21 is the minimum age to purchase a handgun or handgun ammunition under federal law.
    • 21 is the age at which one can purchase multiple tickets to an R-rated film.
    • In some states, 21 is the minimum age to accompany a learner driver, provided that the person supervising the learner has held a full driver license for a specified amount of time. See also: List of minimum driving ages.

In sports

  • Twenty-one is a variation of street basketball, in which each player, of which there can be any number, plays for himself only (i.e. not part of a team); the name comes from the requisite number of baskets.
  • In three-on-three basketball games held under FIBA rules, branded as 3x3, the game ends by rule once either team has reached 21 points.
  • In badminton, and table tennis (before 2001), 21 points are required to win a game.
  • In AFL Women's, the top-level league of women's Australian rules football, each team is allowed a squad of 21 players (16 on the field and five interchanges).

In other fields

See also: List of highways numbered 21
Building called "21" in Zlín, Czech Republic.
Detail of the building entrance

21 is:

  • The Twenty-first Amendment repealed the Eighteenth Amendment, thereby ending Prohibition.
  • The number of spots on a standard cubical (six-sided) die (1+2+3+4+5+6)
  • The number of firings in a 21-gun salute honoring royalty or leaders of countries
  • "Twenty One", a 1994 song by an Irish rock band The Cranberries
  • "21 Guns", a 2009 song by the punk-rock band Green Day
  • Twenty One Pilots, an American musical duo
  • There are 21 trump cards of the tarot deck if one does not consider The Fool to be a proper trump card.
  • The standard TCP/IP port number for FTP connection
  • The Twenty-One Demands were a set of demands which were sent to the China government by the Japanese government of Okuma Shigenobu in 1915
  • 21 Demands of MKS led to the foundation of Solidarity in Poland.
  • In Israel, the number is associated with the profile 21 (the military profile designation granting an exemption from the military service)
  • Duncan MacDougall reported that 21 grams is the weight of the soul, according to an experiment.
  • The number of the French department Côte-d'Or
  • Twenty-One (card game), an ancient card game in which the key value and highest-winning point total is 21
    • Blackjack, a modern version of Twenty-One played in casinos
  • The number of shillings in a guinea.
  • The number of solar rays in the flag of Kurdistan.
  • Twenty-One, an American game show that became the center of the 1950s quiz show scandals when it was shown to be rigged.
  • The number on the logo for the American game show Catch 21
  • The total number of Bitcoin to be released is 21 million “[1]
  • Twenty-One, a 1991 British-American drama film directed by Don Boyd and starring Patsy Kensit.


  1. "Sloane's A000045 : Fibonacci numbers". OEIS Foundation. https://oeis.org/A000045. 
  2. "Sloane's A001006 : Motzkin numbers". OEIS Foundation. https://oeis.org/A001006. 
  3. "Sloane's A000217 : Triangular numbers". OEIS Foundation. https://oeis.org/A000217. 
  4. "Sloane's A000567 : Octagonal numbers". OEIS Foundation. https://oeis.org/A000567. 
  5. "Sloane's A000931 : Padovan sequence". OEIS Foundation. https://oeis.org/A000931. 
  6. "Sloane's A016105 : Blum integers". OEIS Foundation. https://oeis.org/A016105. 
  7. "Sloane's A005349 : Niven (or Harshad) numbers". OEIS Foundation. https://oeis.org/A005349. 
  8. C. J. Bouwkamp, and A. J. W. Duijvestijn, "Catalogue of Simple Perfect Squared Squares of Orders 21 Through 25." Eindhoven University of Technology, Nov. 1992.