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 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, ); 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 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 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:
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
^"Sloane's A000045 : Fibonacci numbers". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2016-05-31.
^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.
^Cohen, Henri (2007). "Consequences of the Hasse–Minkowski Theorem". Number Theory Volume I: Tools and Diophantine Equations. Graduate Texts in Mathematics. Vol. 239 (1st ed.). Springer. pp. 312–314. doi:10.1007/978-0-387-49923-9. ISBN 978-0-387-49922-2. OCLC 493636622. Zbl 1119.11001.
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