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In mathematics 
Thirty-one is the third Mersenne prime ( 25 - 1 ) 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 = 25 - 1 ( 25 - 1). 31 is the eighth Mersenne prime exponent. 31 is also the 4th lucky prime and the 11th supersingular prime.
For the Steiner tree problem, 31 is the number of possible Steiner topologies for Steiner trees with 4 terminals.
No integer added up to its base 10 digits results in 31, making 31 a self number.
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 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 recurrence of the factor 31 in the last number 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. Here, 31 divides every fifteenth number in 3w1 (and 331 every 110th).
In science 
- 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
- The Saros number of the solar eclipse series which began on -1805 January 31 and ended on -489 March 31 (see Astronomical year numbering). The duration of Saros series 31 was 1316.2 years, and it contained 74 solar eclipses.
- The Saros number of the lunar eclipse series which began on -1774 May 30 and ended on -476 July 17. The duration of Saros series 31 was 1298.1 years, and it contained 73 lunar eclipses.
In sports 
The jersey number 31 has been retired by several North American sports teams in honor of past playing greats:
- In Major League Baseball:
- In the NBA:
- In the NHL:
- In the NFL:
In ice hockey goaltenders often wear the number 31.
In other fields 
Thirty-one is also:
- The number of musical triads (12 major, 12 minor, 4 diminished, and 3 augmented). (See triad (music).)
- 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
- A card game (see Thirty-one (game))
- A type of game played on a backgammon board
- The number of flavors of Baskin-Robbins ice cream and 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")
- 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 Christian Huygens and promulgated in the 20th century by Adriaan Fokker
- Room, Adrian. The Guinness Book of Numbers Middlesex: Guinness Publishing Ltd. (1989): 128
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