21 (number)

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

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

Mathematics

Twenty-one is the fifth distinct semiprime,^[1] and the second of the form ${\displaystyle 3\times q}$ where ${\displaystyle q}$ is a higher prime.^[2] It is a repdigit in quaternary (111₄).

Properties

As a

While 21 is the sixth

positive integers

:

$${\displaystyle {\begin{aligned}1&+2+3+4+5+6=21\\1&+(1+2)+(1+3)+(1+2+4)+(1+5)=21\\\end{aligned}}}$$

21 is also the first non-trivial octagonal number.^[5] It is the fifth Motzkin number,^[6] and the seventeenth Padovan number (preceded by the terms 9, 12, and 16, where it is the sum of the first two of these).^[7]

In

${\displaystyle n}$ in decimal such that for any positive integers ${\displaystyle a,b}$ where ${\displaystyle a+b=n}$, at least one of ${\displaystyle {\tfrac {a}{b}}}$ and ${\displaystyle {\tfrac {b}{a}}}$ is a terminating decimal; see proof below:

Proof

For any ${\displaystyle a}$ coprime to ${\displaystyle n}$ and ${\displaystyle n-a}$, the condition above holds when one of ${\displaystyle a}$ and ${\displaystyle n-a}$ only has factors ${\displaystyle 2}$ and ${\displaystyle 5}$ (for a representation in base ten). Let ${\displaystyle A(n)}$ denote the quantity of the numbers smaller than ${\displaystyle n}$ that only have factor ${\displaystyle 2}$ and ${\displaystyle 5}$ and that are coprime to ${\displaystyle n}$, we instantly have ${\displaystyle {\frac {\varphi (n)}{2}}<A(n)}$. We can easily see that for sufficiently large ${\displaystyle n}$, ${\displaystyle A(n)\sim {\frac {\log _{2}(n)\log _{5}(n)}{2}}={\frac {\ln ^{2}(n)}{2\ln(2)\ln(5)}}.}$ However, ${\displaystyle \varphi (n)\sim {\frac {n}{e^{\gamma }\;\ln \ln n}}}$ where ${\displaystyle A(n)=o(\varphi (n))}$ as ${\displaystyle n}$ approaches infinity; thus ${\displaystyle {\frac {\varphi (n)}{2}}<A(n)}$ fails to hold for sufficiently large ${\displaystyle n}$. In fact, for every ${\displaystyle n>2}$, we have ${\displaystyle A(n)<1+\log _{2}(n)+{\frac {3\log _{5}(n)}{2}}+{\frac {\log _{2}(n)\log _{5}(n)}{2}}{\text{ }}}$ 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 ${\displaystyle n>273}$ (actually, when ${\displaystyle n>33}$). Just check a few numbers to see that the complete sequence of numbers having this property is ${\displaystyle \{2,3,4,5,6,7,8,9,11,12,15,21\}.}$

21 is the smallest natural number that is not close to a power of two ${\displaystyle (2^{n})}$, where the range of nearness is ${\displaystyle \pm {n}.}$

Squaring the square

Twenty-one is the smallest number of differently sized squares needed to square the square.^[11]

The lengths of sides of these squares are ${\displaystyle \{2,4,6,7,8,9,11,15,16,17,18,19,24,25,27,29,33,35,37,42,50\}}$ which generate a sum of 427 when excluding a square of side length ${\displaystyle 7}$;^[a] this sum represents the largest square-free integer over a quadratic field of class number two, where 163 is the largest such (Heegner) number of class one.^[12] 427 number is also the first number to hold a sum-of-divisors in equivalence with the third perfect number and thirty-first triangular number (496),^[13][14][15] where it is also the fiftieth number to return ${\displaystyle 0}$ in the Mertens function.^[16]

Quadratic matrices in Z

While the twenty-first prime number 73 is the largest member of Bhargava's definite quadratic 17–integer matrix ${\displaystyle \Phi _{s}(P)}$ representative of all prime numbers,^[17]

$${\displaystyle \Phi _{s}(P)=\{2,3,5,7,11,13,17,19,23,29,31,37,41,43,47,67,{\mathbf {73}}\},}$$

the twenty-first composite number 33 is the largest member of a like definite quadratic 7–integer matrix^[18]

$${\displaystyle \Phi _{s}(2\mathbb {Z} _{\geq 0}+1)=\{1,3,5,7,11,15,{\mathbf {33}}\}}$$

representative of all odd numbers.^[19][b]

In science

• The atomic number of scandium.
• It is very often the day of the solstices in both June and December, though the precise date varies by year.

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.
  • 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

• 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 NASCAR, 21 has been used by Wood Brothers Racing and Ford for decades. The team has won 99 NASCAR Cup Series races, a majority with 21, and 5 Daytona 500's. Their current driver is Harrison Burton.

In other fields

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
  The Fool
  to be a proper trump card.
• The standard
  port number for FTP
  connection
• The
  Okuma Shigenobu
  in 1915
• 21 Demands of MKS led to the foundation of Solidarity
  in Poland.
• In
  profile 21
  (the military profile designation granting an exemption from the military service)
• Duncan MacDougall reported that 21 grams (0.74 oz) is the weight of the soul, according to an experiment
  .
• The number of the French department Côte-d'Or
• Twenty-One
  , 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
• Twenty-One, a 1991 British-American drama film directed by Don Boyd and starring Patsy Kensit
• Cinema XXI (21), A Cinema franchise in Indonesia

References

Wikimedia Commons has media related to 21 (number).