Twenty-one is the fifth distinct
semiprime,[1] and the second of the form where is a higher prime.[2] It is a
repdigit in
quaternary (1114).
Properties
As a
biprime with proper
divisors1,
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).
In
decimal, the number of two-digit prime numbers is twenty-one (a base in which 21 is the fourteenth
Harshad number).[8][9] 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).[10] It is also the largest positive
integer in decimal such that for any positive integers where , at least one of and is a terminating decimal; see proof below:
Proof
For any coprime to and , the condition above holds when one of and only has factors and (for a representation in
base ten).
Let denote the quantity of the numbers smaller than that only have factor and and that are coprime to , we instantly have .
We can easily see that for sufficiently large ,
However, where as approaches
infinity; thus fails to hold for sufficiently large .
In fact, for every , we have
and
So fails to hold when (actually, when ).
Just check a few numbers to see that the complete sequence of numbers having this property is
21 is the smallest natural number that is not close to a
power of two, where the range of nearness is
The lengths of sides of these squares are which generate a sum of
427 when excluding a square of side length ;[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 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 representative of all prime numbers,[17]
the twenty-first composite number
33 is the largest member of a like definite quadratic 7–integer matrix[18]
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.
^This square of side length 7 is adjacent to both the "central square" with side length of 9, and the smallest square of side length 2.
^On the other hand, the largest member of an integer quadratic matrix representative of all numbers is 15,
where the
aliquot sum of 33 is
15, the second such number to have this sum after
16 (
A001065); see also,
15 and 290 theorems. In this sequence, the sum of all members is
^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.