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31 (thirty-one ) is the natural number following 30 and preceding 32 . It is a prime number.
Mathematics
[ edit ]
31 is the 11th prime number. It is a superprime and a self prime (after 3 , 5 , and 7 ), as no integer added up to its base 10 digits results in 31.[1] It is the third Mersenne prime of the form 2n − 1,[2] and the eighth Mersenne prime exponent ,[3] in-turn yielding the maximum positive value for a 32-bit signed binary integer in computing : 2,147,483,647 . After 3 , it is the second Mersenne prime not to be a double Mersenne prime , while the 31st prime number (127 ) is the second double Mersenne prime, following 7 .[4] On the other hand, the thirty-first triangular number is the perfect number 496 , of the form 2(5 − 1) (2 5 − 1) by the Euclid-Euler theorem .[5] 31 is also a primorial prime like its twin prime (29 ),[6] [7] as well as both a lucky prime [8] and a happy number [9] like its dual permutable prime in decimal (13 ).[10]
31 is the number of regular polygons with an odd number of sides that are known to be constructible with compass and straightedge , from combinations of known Fermat primes of the form 22 n + 1 (they are 3 , 5 , 17 , 257 and 65537 ).[11] [12]
31 is a centered pentagonal number .
Only two numbers have a sum-of-divisors equal to 31: 16 (1 + 2 + 4 + 8 + 16) and 25 (1 + 5 + 25), respectively the square of 4 , and of 5 .[13]
31 is the 11th and final consecutive supersingular prime .[14] After 31, the only supersingular primes are 41 , 47 , 59 , and 71 .
31 is the first prime centered pentagonal number ,[15] the fifth centered triangular number ,[16] and the first non-trivial centered decagonal number .[17]
For the Steiner tree problem , 31 is the number of possible Steiner topologies for Steiner trees with 4 terminals.[18]
At 31, the Mertens function sets a new low of −4, a value which is not subceded until 110 .[19]
31 is a repdigit in base 2 (11111) and in base 5 (111).
The cube root of 31 is the value of π correct to four significant figures:
3
3
1
=
3.141
38065
…
{\displaystyle {\sqrt[{3}]{3}}1=3.141\;{\color {red}38065\;\ldots }}
The thirty-first digit in the fractional part of the decimal expansion for pi in base-10 is the last consecutive non-zero digit represented, starting from the beginning of the expansion (i.e, the thirty-second single-digit string is the first
0
{\displaystyle 0}
);[20] the partial sum of digits up to this point is
155
=
31
×
5.
{\displaystyle 155=31\times 5.}
[21] 31 is also the prime partial sum of digits of the decimal expansion of pi after the eighth digit.[22] [a]
The first five Euclid numbers of the form p 1 × p 2 × p 3 × ... × p n + 1 (with p n the n th prime) are prime:[24]
3 = 2 + 1
7 = 2 × 3 + 1
31 = 2 × 3 × 5 + 1
211 = 2 × 3 × 5 × 7 + 1 and
2311 = 2 × 3 × 5 × 7 × 11 + 1
The following term, 30031 = 59 × 509 = 2 × 3 × 5 × 7 × 11 × 13 + 1, is composite .[b] The next prime number of this form has a largest prime p of 31: 2 × 3 × 5 × 7 × 11 × 13 × ... × 31 + 1 ≈ 8.2 × 1033 .[25]
While 13 and 31 in base-ten are the proper first duo of two-digit permutable primes and emirps with distinct digits in base ten, 11 is the only two-digit permutable prime that is its own permutable prime.[10] [26] Meanwhile 1310 in ternary is 1113 and 3110 in quinary is 1115 , with 1310 in quaternary represented as 314 and 3110 as 1334 (their mirror permutations 3314 and 134 , equivalent to 61 and 7 in decimal, respectively, are also prime). (11, 13) form the third twin prime pair[6] between the fifth and sixth prime numbers whose indices add to 11, itself the prime index of 31.[27] Where 31 is the prime index of the fourth Mersenne prime ,[2] the first three Mersenne primes (3 , 7 , 31 ) sum to the thirteenth prime number, 41 .[27] [c] 13 and 31 are also the smallest values to reach record lows in the Mertens function , of −3 and −4 respectively.[29]
The numbers 31, 331, 3331, 33 331 , 333331 , 3 333 331 , and 33 333 331 are all prime. For a time it was thought that every number of the form 3w 1 would be prime. However, the next nine numbers of the sequence are composite; their factorisations are:
333333 331 = 17 × 19 607 843
3 333 333 331 = 673 × 4 952 947
33 333 333 331 = 307 × 108577 633
333333 333 331 = 19 × 83 × 211371 803
3 333 333 333 331 = 523 × 3049 × 2 090 353
33 333 333 333 331 = 607 × 1511 × 1997 × 18 199
333333 333 333 331 = 181 × 1 841 620 626 151
3 333 333 333 333 331 = 199 × 16 750 418 760 469 and
33 333 333 333 333 331 = 31 × 1499 × 717324 094 199 .
The next term (3 17 1 ) is prime, and the recurrence of the factor 31 in the last composite member of the sequence above can be used to prove that no sequence of the type Rw E or ERw can consist only of primes, because every prime in the sequence will periodically divide further numbers.[citation needed ]
31 is the maximum number of areas inside a circle created from the edges and diagonals of an inscribed six-sided polygon , per Moser's circle problem .[30] It is also equal to the sum of the maximum number of areas generated by the first five n -sided polygons: 1, 2, 4, 8, 16, and as such, 31 is the first member that diverges from twice the value of its previous member in the sequence, by 1.
Icosahedral symmetry contains a total of thirty-one axes of symmetry ; six five-fold, ten three-fold, and fifteen two-fold.[31]
In science
[ edit ]
Astronomy
[ edit ]
In sports
[ edit ]
In other fields
[ edit ]
Thirty-one is also:
Notes
[ edit ]
^ On the other hand, "31" as a string represents the first decimal expansion of pi truncated to numbers such that the partial sums of the decimal digits are square numbers .[23]
^ On the other hand, 13 is a largest p of a primorial prime of the form p n # − 1 = 30029 (sequence A057704 in the OEIS ).
^ Also, the sum between the sum and product of the first two Mersenne primes is (3 + 7 ) + (3 × 7) = 10 + 21 = 31 , where its difference (11 ) is the prime index of 31.[27] Thirty-one is also in equivalence with 14 + 17 , which are respectively the seventh composite [28] and prime numbers,[27] whose difference in turn is three .
References
[ edit ]
^ Sloane, N. J. A. (ed.). "Sequence A000043 (Mersenne exponents: primes p such that 2^p - 1 is prime. Then 2^p - 1 is called a Mersenne prime.)" . The On-Line Encyclopedia of Integer Sequences . OEIS Foundation. Retrieved 2023-06-07 .
^ Sloane, N. J. A. (ed.). "Sequence A077586 (Double Mersenne primes)" . The On-Line Encyclopedia of Integer Sequences . OEIS Foundation. Retrieved 2023-06-07 .
^ "Sloane's A000217 : Triangular numbers" . The On-Line Encyclopedia oof Integer Sequences . OEIS Foundation. Retrieved 2022-09-30 .
^ a b Sloane, N. J. A. (ed.). "Sequence A228486 (Near primorial primes: primes p such that p+1 or p-1 is a primorial number (A002110))" . The On-Line Encyclopedia of Integer Sequences . OEIS Foundation. Retrieved 2023-06-07 .
^ Sloane, N. J. A. (ed.). "Sequence A077800 (List of twin primes {p, p+2}.)" . The On-Line Encyclopedia of Integer Sequences . OEIS Foundation. Retrieved 2023-06-07 .
^ "Sloane's A031157 : Numbers that are both lucky and prime" . The On-Line Encyclopedia of Integer Sequences . OEIS Foundation. Retrieved 2016-05-31 .
^ "Sloane's A007770 : Happy numbers" . The On-Line Encyclopedia of Integer Sequences . OEIS Foundation. Retrieved 2016-05-31 .
^ a b Sloane, N. J. A. (ed.). "Sequence A003459 (Absolute primes (or permutable primes): every permutation of the digits is a prime.)" . The On-Line Encyclopedia of Integer Sequences . OEIS Foundation. Retrieved 2023-06-07 .
^ Conway, John H. ; Guy, Richard K. (1996). "The Primacy of Primes" . The Book of Numbers . New York, NY: Copernicus (Springer ). pp. 137–142. doi :10.1007/978-1-4612-4072-3 . ISBN 978-1-4612-8488-8 . OCLC 32854557 . S2CID 115239655 .
^ Sloane, N. J. A. (ed.). "Sequence A004729 (... the 31 regular polygons with an odd number of sides constructible with ruler and compass)" . The On-Line Encyclopedia of Integer Sequences . OEIS Foundation. Retrieved 2023-05-26 .
^ Sloane, N. J. A. (ed.). "Sequence A000203 (The sum of the divisors of n. Also called sigma_1(n ).)" . The On-Line Encyclopedia of Integer Sequences . OEIS Foundation. Retrieved 2024-01-23 .
^ "Sloane's A002267 : The 15 supersingular primes" . The On-Line Encyclopedia of Integer Sequences . OEIS Foundation. Retrieved 2016-05-31 .
^ "Sloane's A005891 : Centered pentagonal numbers" . The On-Line Encyclopedia of Integer Sequences . OEIS Foundation. Retrieved 2016-05-31 .
^ "Sloane's A005448 : Centered triangular numbers" . The On-Line Encyclopedia of Integer Sequences . OEIS Foundation. Retrieved 2016-05-31 .
^ "Sloane's A062786 : Centered 10-gonal numbers" . The On-Line Encyclopedia of Integer Sequences . OEIS Foundation. Retrieved 2016-05-31 .
^ Hwang, Frank. (1992). The Steiner tree problem . Richards , Dana, 1955-, Winter, Pawel, 1952-. Amsterdam: North-Holland. p. 14. ISBN 978-0-444-89098-6 . OCLC 316565524 .
^ Sloane, N. J. A. (ed.). "Sequence A002321 (Mertens's function)" . The On-Line Encyclopedia of Integer Sequences . OEIS Foundation. Retrieved 2023-06-07 .
^ Sloane, N. J. A. (ed.). "Sequence A072136 (Position of the first zero in the fractional part of the base n expansion of Pi.)" . The On-Line Encyclopedia of Integer Sequences . OEIS Foundation. Retrieved 2024-05-30 .
^ Sloane, N. J. A. (ed.). "Sequence A046974 (Partial sums of digits of decimal expansion of Pi.)" . The On-Line Encyclopedia of Integer Sequences . OEIS Foundation.
^ Sloane, N. J. A. (ed.). "Sequence A133213 (Prime partial sums of digits of decimal expansion of pi (A000796).)" . The On-Line Encyclopedia of Integer Sequences . OEIS Foundation. Retrieved 2024-06-02 .
^ Sloane, N. J. A. (ed.). "Sequence A276111 (Decimal expansion of Pi truncated to numbers such that the partial sums of the decimal digits are perfect squares.)" . The On-Line Encyclopedia of Integer Sequences . OEIS Foundation. Retrieved 2024-06-02 .
^ Sloane, N. J. A. (ed.). "Sequence A006862 (Euclid numbers: 1 + product of the first n primes.)" . The On-Line Encyclopedia of Integer Sequences . OEIS Foundation. Retrieved 2023-10-01 .
^ Conway, John H. ; Guy, Richard K. (1996). "The Primacy of Primes" . The Book of Numbers . New York, NY: Copernicus (Springer ). pp. 133–135. doi :10.1007/978-1-4612-4072-3 . ISBN 978-1-4612-8488-8 . OCLC 32854557 . S2CID 115239655 .
^ Sloane, N. J. A. (ed.). "Sequence A006567 (Emirps (primes whose reversal is a different prime).)" . The On-Line Encyclopedia of Integer Sequences . OEIS Foundation. Retrieved 2023-06-16 .
^ a b c d Sloane, N. J. A. (ed.). "Sequence A00040 (The prime numbers.)" . The On-Line Encyclopedia of Integer Sequences . OEIS Foundation. Retrieved 2024-02-09 .
^ Sloane, N. J. A. (ed.). "Sequence A002808 (The composite numbers: numbers n of the form x*y for x greater than 1 and y greater than 1.)" . The On-Line Encyclopedia of Integer Sequences . OEIS Foundation. Retrieved 2024-02-10 .
^ Sloane, N. J. A. (ed.). "Sequence A051402 (Inverse Mertens function: smallest k such that |M(k )| is n, where M(x ) is Mertens's function A002321.)" . The On-Line Encyclopedia of Integer Sequences . OEIS Foundation. Retrieved 2024-02-08 .
^ "Sloane's A000127 : Maximal number of regions obtained by joining n points around a circle by straight lines" . The On-Line Encyclopedia of Integer Sequences . OEIS Foundation. Retrieved 2022-09-30 .
^ Hart, George W. (1998). "Icosahedral Constructions" (PDF) . In Sarhangi, Reza (ed.). Bridges: Mathematical Connections in Art, Music, and Science . Proceedings of the Bridges Conference . Winfield, Kansas. p. 196. ISBN 978-0966520101 . OCLC 59580549 . S2CID 202679388 .{{cite book }}
: CS1 maint: location missing publisher (link )
^ "Tureng - 31 çekmek - Türkçe İngilizce Sözlük" . tureng.com . Retrieved 2023-01-18 .
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