# 21 (number)

21 (twenty-one) is the natural number following 20 and preceding 22.

 ← 20 21 22 →
Cardinaltwenty-one
Ordinal21st
(twenty-first)
Factorization3 × 7
Divisors1, 3, 7, 21
Greek numeralΚΑ´
Roman numeralXXI
Binary101012
Ternary2103
Quaternary1114
Quinary415
Senary336
Octal258
Duodecimal1912
Vigesimal1120
Base 36L36

## In mathematics

21 is:

• a Blum integer, since it is a semiprime with both its prime factors being Gaussian primes.[1]
• a Fibonacci number.[2]
• a Motzkin number.[4]
• a triangular number,[5] because it is the sum of the first six natural numbers (1 + 2 + 3 + 4 + 5 + 6 = 21).
• an octagonal number.[6]
• a composite number, its proper divisors being 1, 3 and 7.
• the sum of the divisors of the first 5 positive integers.
• the smallest non-trivial example of a Fibonacci number whose digits are Fibonacci numbers and whose digit sum is also a Fibonacci number.
• a repdigit in base 4 (1114).
• the smallest natural number that is not close to a power of 2, 2n, where the range of closeness is ±n.
• the smallest number of differently sized squares needed to square the square.[7]
• the largest n with this property: for any positive integers a,b such that a + b = n, at least one of ${\displaystyle {\tfrac {a}{b}}}$  and ${\displaystyle {\tfrac {b}{a}}}$  is a terminating decimal. See a brief proof below.

Note that a necessary condition for n is that for any a coprime to n, a and n - a must satisfy the condition above, therefore at least one of a and n - a must only have factor 2 and 5.

Let ${\displaystyle A(n)}$  donate the quantity of the numbers smaller than n that only have factor 2 and 5 and that are coprime to n, we instantly have ${\displaystyle {\frac {\varphi (n)}{2}} .

We can easily see that for sufficiently large n, ${\displaystyle A(n)\sim {\frac {\log _{2}(n)\log _{5}(n)}{2}}={\frac {\ln ^{2}(n)}{2\ln(2)\ln(5)}}}$ , but ${\displaystyle \varphi (n)\sim {\frac {n}{e^{\gamma }\;\ln \ln n}}}$ , ${\displaystyle A(n)=o(\varphi (n))}$  as n goes to infinity, thus ${\displaystyle {\frac {\varphi (n)}{2}}  fails to hold for sufficiently large n.

In fact, For every n > 2, we have

${\displaystyle A(n)<1+\log _{2}(n)+{\frac {3\log _{5}(n)}{2}}+{\frac {\log _{2}(n)\log _{5}(n)}{2}}}$

and

${\displaystyle \varphi (n)>{\frac {n}{e^{\gamma }\;\log \log n+{\frac {3}{\log \log n}}}}}$

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

Just check a few numbers to see that n = 2, 3, 4, 5, 6, 7, 8, 9, 11, 12, 15, 21.

21 appears in the Padovan sequence, preceded by the terms 9, 12, 16 (it is the sum of the first two of these).[8]

## Age 21

• In several countries 21 is the age of majority. See also: Coming of age.
• In all US states, 21 is the drinking age.
• However, in Puerto Rico and U.S. Virgin Islands, the drinking age is 18.
• In California, Hawaii, New York, and New Jersey, 21 is the minimum age that one person may purchase cigarettes and other tobacco products.
• In some countries it is the voting age.
• In the United States, 21 is the age at which one can purchase multiple tickets to an R-rated film without providing identification. It is also the age to accompany one under the age of 17 as their parent or adult guardian for an R-rated movie.
• In most US states, 21 is the minimum age at which a person may gamble or enter casinos.
• In 2011, Adele named her second studio album 21, because of her age at the time.
• In the United Kingdom, 21 is the minimum age to accompany a learner driver, provided that the person supervising the learner has held a full driving licence for a minimum of three years.

## In other fields

Building called "21" in Zlín, Czech Republic.

Detail of the building entrance

21 is:

## References

1. ^ "Sloane's A016105 : Blum integers". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2016-05-31.
2. ^ "Sloane's A000045 : Fibonacci numbers". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2016-05-31.
3. ^ "Sloane's A005349 : Niven (or Harshad) numbers". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2016-05-31.
4. ^ "Sloane's A001006 : Motzkin numbers". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2016-05-31.
5. ^ "Sloane's A000217 : Triangular numbers". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2016-05-31.
6. ^ "Sloane's A000567 : Octagonal numbers". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2016-05-31.
7. ^ 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.
8. ^ "Sloane's A000931 : Padovan sequence". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2016-05-31.