← 30 32 → 31
← 30 31 32 33 34 35 36 37 38 39 → List of numbers — Integers 0 10 20 30 40 50 60 70 80 90 →
Cardinal	thirty-one
Ordinal	31st (thirty-first)
Factorization	prime
Divisors	1, 31
Roman numeral	XXXI
Binary	111112
Octal	378
Duodecimal	2712
Hexadecimal	1F16

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

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

Contents 1 In mathematics 2 In science 2.1 Astronomy 3 In sports 4 In other fields 5 Historical years 6 References

[edit] 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 = 2^{5 - 1} ( 2⁵ - 1). 31 is the eighth Mersenne prime exponent. 31 is also the 4th lucky prime and the 11th supersingular prime.

31 is a centered triangular number, a centered pentagonal number and centered decagonal number.

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.

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_w1 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
• 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_wE 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_w1 (and 331 every 110th).

[edit] In science

• The atomic number of gallium

[edit] 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
• The Saros number of the solar eclipse series which began on -1805 January 31 and ended on -489 March 31. 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.

[edit] 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:
  • The San Diego Padres, for Hall of Famer Dave Winfield.
  • The Chicago Cubs, for Hall of Famer Ferguson Jenkins and likely future Hall of Famer Greg Maddux.
  • The Atlanta Braves, also for Maddux.
• In the NBA:
  • The Boston Celtics, for Cedric Maxwell.
  • The Indiana Pacers, for Reggie Miller.
• In the NHL:
  • The Edmonton Oilers, for Grant Fuhr.
  • The New York Islanders, for Billy Smith.
• In the NFL:
  • The Atlanta Falcons, for William Andrews.
  • The New Orleans Saints, for Jim Taylor.

NASCAR driver Jeff Burton drives #31, a car which was subject to a controversy when one of the sponsors changed its name after merging with another company.

Ice hockey goaltenders (typically backups) use 31 when 30 is taken.

[edit] 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
• A type of card game (see Thirty-one (game))^[1]
• 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 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

[edit] Historical years

31 A.D., 31 B.C., 1931, 2031, etc.

[edit] References

• Prime Curios! 31 from the Prime Pages