31 (number)

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

Template:Numbers 30s
Cardinal	31 thirty-one
Ordinal	31st thirty-first
Factorization	prime
Divisors	1, 31
Roman numeral	XXXI
Binary	11111
Hexadecimal	1F

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).

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
• 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.

In sports

• The shirt number of Bayern Munich midfielder Bastian Schweinsteiger
• The jersey number of basketball great and retired Indiana Pacer Reggie Miller
• The jersey number of famed NHL goaltender Curtis "CuJo" Joseph, who currently plays for the Phoenix Coyotes
• Is the jersey number of Baseball Hall of Famer Ferguson Jenkins and Greg Maddux
• Is the retired jersey number of Baseball Hall of Famer Dave Winfield
• The jersey number of Hall of Fame NHL Goaltender Grant Fuhr
• the jersey number of Phoenix Suns forward Shawn Marion
• The jersey number for Cleveland Browns Running Back Jamal Lewis
• The car number of NASCAR driver Jeff Burton
• The jersey number of San Diego Chargers' defensive back Antonio Cromartie.
• The jersey number of Dallas Cowboys safety Roy Williams.

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))
• 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
• In Star Trek, the name of Section 31, a secret police shadow organization of the Federation
• 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)

Historical years

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

References

• Prime Curios! 31 from the Prime Pages