31 (number)

Short description: Natural number
 ← 30 31 32 →
Cardinalthirty-one
Ordinal31st
(thirty-first)
Factorizationprime
Prime11th
Divisors1, 31
Greek numeralΛΑ´
Roman numeralXXXI
Binary111112
Ternary10113
Quaternary1334
Quinary1115
Senary516
Octal378
Duodecimal2712
Vigesimal1B20
Base 36V36

31 (thirty-one) is the natural number following 30 and preceding 32. It is a prime number.

In 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 and 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:

$\displaystyle{ \sqrt[3]31 = 3.141\;{\color{red}38065\;\ldots} }$

The first five Euclid numbers of the form p1 × p2 × p3 × ... × pn + 1 (with pn the nth 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.[lower-alpha 1] 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) are the third twin prime pair,[6] formed by the fifth and sixth prime numbers, whose indices add to 11, itself the prime index of 31.

The numbers 31, 331, 3331, 33331, 333331, 3333331, and 33333331 are all prime. For a time it was thought that every number of the form 3w1 would be prime. However, the next nine numbers of the sequence are composite; their factorisations are:

• 333333331 = 17 × 19607843
• 3333333331 = 673 × 4952947
• 33333333331 = 307 × 108577633
• 333333333331 = 19 × 83 × 211371803
• 3333333333331 = 523 × 3049 × 2090353
• 33333333333331 = 607 × 1511 × 1997 × 18199
• 333333333333331 = 181 × 1841620626151
• 3333333333333331 = 199 × 16750418760469 and
• 33333333333333331 = 31 × 1499 × 717324094199.

The next term (3171) 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 RwE 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.[23] 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.[24]

In other fields

Thirty-one is also:

• The number of days in each of the months January, March, May, July, August, October and December
• The number of the date that Halloween and New Year's Eve are celebrated
• The code for international direct-dial phone calls to the Netherlands
• Thirty-one, a card game
• The number of kings defeated by the incoming Israelites in Canaan according to Joshua 12:24: "all the kings, one and thirty" (Wycliffe Bible translation)
• A type of game played on a backgammon board
• The number of flavors of Baskin-Robbins ice cream; the shops are called 31 Ice Cream in Japan
• ISO 31 is the ISO's standard for quantities and units
• In the title of the anime Ulysses 31
• In the title of Nick Hornby's book 31 Songs
• A women's honorary at The University of Alabama (XXXI)
• The number of the French department Haute-Garonne
• In music, 31-tone equal temperament is a historically significant tuning system (31 equal temperament), first theorized by Christiaan Huygens and promulgated in the 20th century by Adriaan Fokker
• Number of letters in Macedonian alphabet
• Number of letters in Ottoman alphabet
• The number of years approximately equal to 1 billion seconds
• A slang term for masturbation in Turkish.[25]

References

1. On the other hand, 13 is a largest p of a primorial prime of the form pn# − 1 = 30029 (sequence A057704 in the OEIS).
1. "Sloane's A003052 : Self numbers". OEIS Foundation.
2. Sloane, N. J. A., ed. "Sequence A000668 (Mersenne primes (primes of the form 2^n - 1).)". OEIS Foundation. Retrieved 2023-06-07.
3. 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.)". OEIS Foundation. Retrieved 2023-06-07.
4. Sloane, N. J. A., ed. "Sequence A077586 (Double Mersenne primes)". OEIS Foundation. Retrieved 2023-06-07.
5. "Sloane's A000217 : Triangular numbers". OEIS Foundation.
6. Sloane, N. J. A., ed. "Sequence A228486 (Near primorial primes: primes p such that p+1 or p-1 is a primorial number (A002110))". OEIS Foundation. Retrieved 2023-06-07.
7. Sloane, N. J. A., ed. "Sequence A077800 (List of twin primes {p, p+2}.)". OEIS Foundation. Retrieved 2023-06-07.
8. "Sloane's A007770 : Happy numbers". OEIS Foundation.
9. Sloane, N. J. A., ed. "Sequence A003459 (Absolute primes (or permutable primes): every permutation of the digits is a prime.)". OEIS Foundation. Retrieved 2023-06-07.
10. 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.
11. Sloane, N. J. A., ed. "Sequence A004729 (... the 31 regular polygons with an odd number of sides constructible with ruler and compass)". OEIS Foundation. Retrieved 2023-05-26.
12. Sloane, N. J. A., ed. "Sequence A000203 (The sum of the divisors of n. Also called sigma_1(n).)". OEIS Foundation. Retrieved 2024-01-23.
13. "Sloane's A002267 : The 15 supersingular primes". OEIS Foundation.
14. "Sloane's A005891 : Centered pentagonal numbers". OEIS Foundation.
15. "Sloane's A005448 : Centered triangular numbers". OEIS Foundation.
16. "Sloane's A062786 : Centered 10-gonal numbers". OEIS Foundation.
17. Hwang, Frank. (1992). The Steiner tree problem. Richards, Dana, 1955-, Winter, Pawel, 1952-. Amsterdam: North-Holland. pp. 14. ISBN 978-0-444-89098-6. OCLC 316565524.
18. Sloane, N. J. A., ed. "Sequence A002321 (Mertens's function)". OEIS Foundation. Retrieved 2023-06-07.
19. Sloane, N. J. A., ed. "Sequence A006862 (Euclid numbers: 1 + product of the first n primes.)". OEIS Foundation. Retrieved 2023-10-01.
20. 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.
21. Sloane, N. J. A., ed. "Sequence A006567 (Emirps (primes whose reversal is a different prime).)". OEIS Foundation. Retrieved 2023-06-16.
22. Sarhangi, Reza, ed (1998). "Icosahedral Constructions". Bridges: Mathematical Connections in Art, Music, and Science. Proceedings of the Bridges Conference. Winfield, Kansas. p. 196. ISBN 978-0966520101. OCLC 59580549.