21 (واحد وعشرون) هو عدد صحيح يلي العدد 20 ويسبق العدد 22 وهو عدد طبيعي موجب.

← 20 21 22 →
كميخطأ: الوظيفة "numeral_to_arabic" غير موجودة.
ترتيبي21st
(خطأ: الوظيفة "numeral_to_arabic" غير موجودة.)
التحليل لعوامل3 × 7
القواسم1, 3, 7, 21
العدد اليونانيΚΑ´
العدد الرومانيXXI
ثنائي101012
ثلاثي2103
رباعي1114
خماسي415
سداسي336
ثماني258
اثنا عشري1912
ستة عشري1516
عشريني1120
أساس 36L36


القرن الحالي هو القرن 21 الميلادي، في التقويم الگريگوري.

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الرياضيات

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).


الخصائص

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 two discrete semiprimes (21, 22), where the next such cluster is (38, 39).

21 is the first Blum integer, since it is a semiprime with both its prime factors being Gaussian primes.[3]

While 21 is the sixth triangular number,[4] it is also the sum of the divisors of the first five positive integers:

 

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:

البرهان

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  

تربيع المربع

 
The minimum number of squares needed to square the square (using different edge-lengths) is 21.

Twenty-one is the smallest number of differently sized squares needed to square the square.[11]

The lengths of sides of these squares are   which generate a sum of 427 when excluding a square of side length  ;[أ] 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]

 

representative of all odd numbers.[19][ب]

في العلوم

العمر 21

  • In thirteen countries, 21 is the age of majority. See also: Coming of age.
  • In eight countries, 21 is the minimum age to purchase tobacco products.
  • In seventeen countries, 21 is the drinking age.
  • In nine countries, it is the voting age.
  • In the United States:
    • 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.

في الرياضة

  • 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.
  • In badminton, and table tennis (before 2001), 21 points are required to win a game.
  • In AFL Women's, the top-level league of women's Australian rules football, each team is allowed a squad of 21 players (16 on the field and five interchanges).
  • In NASCAR, 21 has been used by Wood Brothers Racing and Ford for decades. The team has won 99 NASCAR Cup Series races, a majority with 21, and 5 Daytona 500's. Their current driver is Harrison Burton.


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في المجالات الأخرى

 
Building called "21" in Zlín, Czech Republic
 
Detail of the building entrance

21 هو:

ملاحظات

  1. ^ 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.
  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  

المراجع

  1. ^ قالب:Cite OEIS
  2. ^ قالب:Cite OEIS
  3. ^ "Sloane's A016105 : Blum integers". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2016-05-31.
  4. ^ "Sloane's A000217 : Triangular numbers". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2016-05-31.
  5. ^ "Sloane's A000567 : Octagonal numbers". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2016-05-31.
  6. ^ "Sloane's A001006 : Motzkin numbers". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2016-05-31.
  7. ^ "Sloane's A000931 : Padovan sequence". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2016-05-31.
  8. ^ "Sloane's A005349 : Niven (or Harshad) numbers". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2016-05-31.
  9. ^ قالب:Cite OEIS
  10. ^ "Sloane's A000045 : Fibonacci numbers". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2016-05-31.
  11. ^ 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.
  12. ^ قالب:Cite OEIS
  13. ^ قالب:Cite OEIS
  14. ^ قالب:Cite OEIS
  15. ^ قالب:Cite OEIS
  16. ^ قالب:Cite OEIS
  17. ^ قالب:Cite OEIS
  18. ^ قالب:Cite OEIS
  19. ^ 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.