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, and the second of the form $$3 \times q$$ where $$q$$ is a higher prime. It is a repdigit in quaternary (1114).

Properties
As a biprime with proper divisors 1, 3 and 7, twenty-one has a prime aliquot sum of 11 within an aliquot sequence containing only one composite number (21, 11, 1, 0); it is the second composite number with an aliquot sum of 11, following 18. 21 is the first member of the second cluster of consecutive discrete semiprimes (21, 22), where the next such cluster is (33, 34, 35).

21 is the first Blum integer, since it is a semiprime with both its prime factors being Gaussian primes.

While 21 is the sixth triangular number, it is also the sum of the divisors of the first five positive integers:

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

21 is also the first non-trivial octagonal number. It is the fifth Motzkin number, and the seventeenth Padovan number (preceded by the terms 9, 12, and 16, where it is the sum of the first two of these). In decimal, the number of two-digit prime numbers is twenty-one (a base in which 21 is the fourteenth Harshad number). It is the smallest non-trivial example in base ten of a Fibonacci number (where 21 is the 8th member, as the sum of the preceding terms in the sequence 8 and 13) whose digits (2, 1) are Fibonacci numbers and whose digit sum is also a Fibonacci number (3). It is also the largest positive integer $$n$$ in decimal such that for any positive integers $$a,b$$ where $$a + b = n$$, at least one of $$\tfrac{a}{b}$$ and $$\tfrac{b}{a}$$ is a terminating decimal; see proof below:

For any $$a$$ coprime to $$n$$ and $$n - a$$, the condition above holds when one of $$a$$ and $$n - a$$ only has factors $$2$$ and $$5$$ (for a representation in base ten).

Let $$A(n)$$ 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 $$\frac{\varphi(n)}{2} < A(n)$$.

We can easily see that for sufficiently large $$n$$, $$A(n) \sim \frac{\log_2(n) \log_5(n)}{2} = \frac{\ln^{2}(n)}{2 \ln(2) \ln(5)}.$$

However, $$\varphi(n) \sim \frac {n} {e^\gamma\; \ln \ln n}$$ where $$A(n) = o(\varphi(n))$$ as $$n$$ approaches infinity; thus $$\frac{\varphi(n)}{2} < A(n)$$ fails to hold for sufficiently large $$n$$.

In fact, for every $$n > 2$$, we have
 * $$A(n)< 1 + \log_2(n) + \frac{3 \log_5(n)}{2} + \frac{\log_2(n) \log_5(n)}{2} \text { }$$ and
 * $$\varphi(n) > \frac {n} {e^\gamma\; \log \log n + \frac {3} {\log \log n}}.$$

So $$\frac{\varphi(n)}{2} < A(n)$$ fails to hold when $$n > 273$$ (actually, when $$n > 33$$).

Just check a few numbers to see that the complete sequence of numbers having this property is $$\{2, 3, 4, 5, 6, 7, 8, 9, 11, 12, 15, 21\}.$$

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

Squaring the square


Twenty-one is the smallest number of differently sized squares needed to square the square.

The lengths of sides of these squares are $$\{2,4,6,7,8,9,11,15,16,17,18,19,24,25,27,29,33,35,37,42,50\}$$ which generate a sum of 427 when excluding a square of side length $$7$$; 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. 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),  where it is also the fiftieth number to return $$0$$ in the Mertens function.

Quadratic matrices in Z
While the twenty-first prime number 73 is the largest member of Bhargava's definite quadratic 17–integer matrix $$\Phi_{s}(P)$$ representative of all prime numbers, $$\Phi_{s}(P) = \{2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 67, \bold {73} \},$$

the twenty-first composite number 33 is the largest member of a like definite quadratic 7–integer matrix $$\Phi_{s}(2\mathbb {Z} _{\ge 0} + 1) = \{1, 3, 5, 7, 11, 15, \bold {33} \}$$

representative of all odd numbers.

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

 * 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 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 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 g 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