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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.

## Contents

## In mathematics

31 is the number of regular polygons with an odd number of sides that are known to be constructible with compass and straightedge.

31 is the third Mersenne prime (2^{5} − 1)^{[1]} and the eighth Mersenne prime exponent, as well as the fourth primorial prime, and together with twenty-nine, another primorial prime, it comprises a twin prime. As a Mersenne prime, 31 is related to the perfect number 496, since 496 = 2^{(5 − 1)}(2^{5} − 1). 31 is also the 4th lucky prime^{[2]} and the 11th supersingular prime.^{[3]}

31 is a centered triangular number,^{[4]} the lowest prime centered pentagonal number^{[5]} and a centered decagonal number.^{[6]}

For the Steiner tree problem, 31 is the number of possible Steiner topologies for Steiner trees with 4 terminals.

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

No integer added up to its base 10 digits results in 31, making 31 a self number.^{[7]}

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

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).

## 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.
- There are 31 teams in the National Hockey League.

## In other fields

**Thirty-one** is also:

- The number of days in the months January, March, May, July, August, October and December
- 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* - Turkish slang for male masturbation ("otuzbir")
^{[citation needed]} - 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

## References

**^**"Sloane's A000668 : Mersenne primes".*The On-Line Encyclopedia of Integer Sequences*. OEIS Foundation. Retrieved 2016-05-31.**^**"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 A002267 : The 15 supersingular primes".*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 A005891 : Centered pentagonal 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.**^**"Sloane's A003052 : Self numbers".*The On-Line Encyclopedia of Integer Sequences*. OEIS Foundation. Retrieved 2016-05-31.

## External links

Look up in Wiktionary, the free dictionary.thirty-one |

Wikimedia Commons has media related to .31 (number) |

- Prime Curios! 31 from the Prime Pages