21 (number)

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

← 20 21 22 →
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Cardinal	twenty-one
Ordinal	21st (twenty-first)
Factorization	3 × 7
Divisors	1, 3, 7, 21
Greek numeral	ΚΑ´
Roman numeral	XXI
Binary	101012
Ternary	2103
Quaternary	1114
Quinary	415
Senary	336
Octal	258
Duodecimal	1912
Hexadecimal	1516
Vigesimal	1120
Base 36	L36

In mathematics

21 is:

• a Blum integer, since it is a semiprime with both its prime factors being Gaussian primes.[1]
• a Fibonacci number.[2]
• a Harshad number.[3]
• a Motzkin number.[4]
• a triangular number,[5] because it is the sum of the first six natural numbers (1 + 2 + 3 + 4 + 5 + 6 = 21).
• an octagonal number.[6]
• a composite number, its proper divisors being 1, 3 and 7.
• the sum of the divisors of the first 5 positive integers.
• the smallest non-trivial example of a Fibonacci number whose digits are Fibonacci numbers and whose digit sum is also a Fibonacci number.
• 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.[7]
• the largest n with this property: for any positive integers a,b such that a + b = n, at least one of ${\displaystyle {\tfrac {a}{b}}}$ and ${\displaystyle {\tfrac {b}{a}}}$ 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 ${\displaystyle A(n)}$ donate the quantity of the numbers smaller than n that only have factor 2 and 5 and that are coprime to n, we instantly have ${\displaystyle {\frac {\varphi (n)}{2}}<A(n)}$. We can easily see that for sufficiently large n, ${\displaystyle A(n)\sim {\frac {\log _{2}(n)\log _{5}(n)}{2}}={\frac {\ln ^{2}(n)}{2\ln(2)\ln(5)}}}$, but ${\displaystyle \varphi (n)\sim {\frac {n}{e^{\gamma }\;\ln \ln n}}}$, ${\displaystyle A(n)=o(\varphi (n))}$ as n goes to infinity, thus ${\displaystyle {\frac {\varphi (n)}{2}}<A(n)}$ fails to hold for sufficiently large n. In fact, For every n > 2, we have ${\displaystyle A(n)<1+\log _{2}(n)+{\frac {3\log _{5}(n)}{2}}+{\frac {\log _{2}(n)\log _{5}(n)}{2}}}$ and ${\displaystyle \varphi (n)>{\frac {n}{e^{\gamma }\;\log \log n+{\frac {3}{\log \log n}}}}}$ so ${\displaystyle {\frac {\varphi (n)}{2}}<A(n)}$ fails to hold when n > 273 (actually, when n > 33). Just check a few numbers to see that n = 2, 3, 4, 5, 6, 7, 8, 9, 11, 12, 15, 21.

21 appears in the Padovan sequence, preceded by the terms 9, 12, 16 (it is the sum of the first two of these).[8]

In science

• The atomic number of scandium.

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 most US states, 21 is the minimum age at which a person may gamble or enter casinos.
• In the United States, 21 is the minimum age to purchase a handgun or handgun ammunition.
• In the United States, 21 is the age at which one can purchase multiple tickets to an R-rated film without providing identification. It is also the age to accompany one under the age of 17 as their parent or adult guardian for an R-rated movie.
• In some countries, 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 2011, Adele named her second studio album 21, because of her age at the time.

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
• 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 Chinese 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.
• The number on the logo for the American game show Catch 21

References

1. "Sloane's A016105 : Blum integers". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2016-05-31.
2. "Sloane's A000045 : Fibonacci numbers". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2016-05-31.
3. "Sloane's A005349 : Niven (or Harshad) numbers". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2016-05-31.
4. "Sloane's A001006 : Motzkin numbers". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2016-05-31.
5. "Sloane's A000217 : Triangular numbers". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2016-05-31.
6. "Sloane's A000567 : Octagonal numbers". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2016-05-31.
7. 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.
8. "Sloane's A000931 : Padovan sequence". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2016-05-31.