Natural number
31 (thirty-one ) is the natural number following
30 and preceding
32 . It is a prime number.
Mathematics
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) (25 − 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 22n + 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 a
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 first five
Euclid numbers of the form p1 × p2 × p3 × ... × pn + 1 (with pn the n th prime) are prime:
[20]
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 .
[a] 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 .
[21]
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]
[22] 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.
[23] 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 .
[23]
[b] 13 and 31 are also the smallest values to reach record lows in the
Mertens function , of −3 and −4 respectively.
[25]
The numbers 31, 331, 3331, 33331 , 333331 , 3333 331 , and 33333 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 × 19607 843
3333 333 331 = 673 × 4952 947
33333 333 331 = 307 × 108577 633
333333 333 331 =
19 ×
83 × 211371 803
3333 333 333 331 = 523 × 3049 × 2090 353
33333 333 333 331 = 607 × 1511 × 1997 × 18199
333333 333 333 331 =
181 × 1841 620 626 151
3333 333 333 333 331 =
199 × 16750 418 760 469 and
33333 333 333 333 331 = 31 × 1499 × 717324 094 199 .
The next term (317 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 .
[26] 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.
[27]
In science
Astronomy
In sports
In other fields
Thirty-one is also:
Notes
^ On the other hand,
13 is a largest p of a
primorial prime of the form pn # − 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.
[23] Thirty-one is also in equivalence with 14 + 17 , which are respectively the seventh
composite
[24] and prime numbers,
[23] whose difference in turn is three .
References
^
"Sloane's A003052 : Self numbers" . The On-Line Encyclopedia of Integer Sequences . OEIS Foundation. Retrieved 2016-05-31 .
^
a
b
Sloane, N. J. A. (ed.).
"Sequence A000668 (Mersenne primes (primes of the form 2^n - 1).)" . The
On-Line Encyclopedia of Integer Sequences . OEIS Foundation. Retrieved 2023-06-07 .
^
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 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 .
External links
Look up
thirty-one in Wiktionary, the free dictionary.
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