# 31 (number)

__: Natural number__

**Short description**
| ||||
---|---|---|---|---|

Cardinal | thirty-one | |||

Ordinal | 31st (thirty-first) | |||

Factorization | prime | |||

Prime | 11th | |||

Divisors | 1, 31 | |||

Greek numeral | ΛΑ´ | |||

Roman numeral | XXXI | |||

Binary | 11111_{2} | |||

Ternary | 1011_{3} | |||

Quaternary | 133_{4} | |||

Quinary | 111_{5} | |||

Senary | 51_{6} | |||

Octal | 37_{8} | |||

Duodecimal | 27_{12} | |||

Hexadecimal | 1F_{16} | |||

Vigesimal | 1B_{20} | |||

Base 36 | V_{36} |

**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, and twin prime with 29. It is a self prime (after 3, 5, and 7) as no integer added up to its base 10 digits results in 31.^{[1]} It is also a lucky prime^{[2]}, a primorial prime, and a happy number.^{[3]}

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 2^{2n} + 1.

31 is the third Mersenne prime of the form 2^{n} − 1. It is also the eighth Mersenne prime exponent, specifically for the number 2,147,483,647, which is the maximum positive value for a 32-bit signed binary integer in computing. After 3, it is the second Mersenne prime not to be a double Mersenne prime. 127, which is the 31^{st} prime number, is a double Mersenne prime. The 31^{st} triangular number is the perfect number 496, of the form 2^{(5 − 1)}(2^{5} − 1).^{[4]}

31 is a centered triangular number,^{[5]} the first prime centered pentagonal number^{[6]} and a centered decagonal number.^{[7]}

For the Steiner tree problem, 31 is the number of possible Steiner topologies for Steiner trees with 4 terminals.^{[8]}

At 31, the Mertens function sets a new low of −4, a value which is not subceded until 110.

31 is a repdigit in base 5 (111), and base 2 (11111).

The cube root of 31 is the value of pi correct to four significant figures.

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 3_{w}1 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 recurrence of the factor 31 in the last number above can be used to prove that no sequence of the type R_{w}E or ER_{w} can consist only of primes because every prime in the sequence will periodically divide further numbers. Here, 31 divides every fifteenth number in 3_{w}1 (and 331 every 110th).

31 is the 11th and final consecutive supersingular prime.^{[9]} After 31, the only supersingular primes are 41, 47, 59, and 71.

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.^{[10]} 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.

## In science

- The atomic number of gallium

### Astronomy

- Messier object M31, a magnitude 4.5 galaxy in the constellation Andromeda. It is also known as the Andromeda Galaxy, and is readily visible to the naked eye in a modestly dark sky.
- The New General Catalogue object NGC 31, a spiral galaxy in the constellation Phoenix

## In sports

- Ice hockey goaltenders often wear the number 31.

## In other fields

**Thirty-one** is also:

- The number of days in 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 Israelite settlers 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

## References

- ↑ "Sloane's A003052 : Self numbers". OEIS Foundation. https://oeis.org/A003052.
- ↑ "Sloane's A031157 : Numbers that are both lucky and prime". OEIS Foundation. https://oeis.org/A031157.
- ↑ "Sloane's A007770 : Happy numbers". OEIS Foundation. https://oeis.org/A007770.
- ↑ "Sloane's A000217 : Triangular numbers". OEIS Foundation. https://oeis.org/A000217.
- ↑ "Sloane's A005448 : Centered triangular numbers". OEIS Foundation. https://oeis.org/A005448.
- ↑ "Sloane's A005891 : Centered pentagonal numbers". OEIS Foundation. https://oeis.org/A005891.
- ↑ "Sloane's A062786 : Centered 10-gonal numbers". OEIS Foundation. https://oeis.org/A062786.
- ↑ 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. https://www.worldcat.org/oclc/316565524. - ↑ "Sloane's A002267 : The 15 supersingular primes". OEIS Foundation. https://oeis.org/A002267.
- ↑ "Sloane's A000127 : Maximal number of regions obtained by joining n points around a circle by straight lines.". OEIS Foundation. https://oeis.org/A000127.

## External links

- Prime Curios! 31 from the Prime Pages

Original source: https://en.wikipedia.org/wiki/31 (number).
Read more |