Twenty-one is the fifth distinctsemiprime,[1] and the second of the form where is a higher prime.[2] It is arepdigit inquaternary (1114).
Properties
As abiprime with properdivisors1,3 and7, twenty-one has a primealiquot sum of11 within an aliquot sequence containing only one composite number (21,11,1,0). 21 is the first member of the second cluster of consecutive discretesemiprimes (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 between100 and200.
Indecimal, the number of two-digit prime numbers is twenty-one (a base in which 21 is the fourteenthHarshad number).[8][9] It is the smallest non-trivial example in base ten of aFibonacci number (where 21 is the 8th member, as the sum of the preceding terms in the sequence8 and13) whose digits (2,1) are Fibonacci numbers and whosedigit sum is also a Fibonacci number (3).[10] It is also the largest positiveinteger 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 inbase 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 approachesinfinity; 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 apower of two, where the range of nearness is
The lengths of sides of these squares are which generate a sum of427 when excluding a square of side length;[a] this sum represents the largest square-free integer over a quadratic field of class number two, where163 is the largest such (Heegner) number of class one.[12] 427 number is also the first number to hold asum-of-divisors in equivalence with the thirdperfect number and thirty-firsttriangular number (496),[13][14][15] where it is also the fiftieth number to return in theMertens function.[16]
21 is theminimum age at which a person maygamble or entercasinos in most states (since alcohol is usually provided).
21 is the minimum age to purchase ahandgun 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.
^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 ofall numbers is 15, where thealiquot sum of 33 is15, the second such number to have this sum after16 (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.
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