21 (twenty-one) is 9 plus 10 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). There are 21 prime numbers with 2 digits. There are A total of 21 prime numbers between 100 and 200. 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: 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](https://bliptext.com/articles/[1](https://bliptext.com/articles/1)[3](https://bliptext.com/articles/3)-number)) 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 and 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. We can easily see that for sufficiently large n, However, where as n approaches infinity; thus fails to hold for sufficiently large n. In fact, for every n > 2, we have So 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 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 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, the twenty-first composite number 33 is the largest member of a like definite quadratic 7–integer matrix representative of all odd numbers.

In science

Age 21

In sports

In other fields

21 is: