31 (number)

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

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 jersey number 31 was retired for Baseball Hall of Famer Dave Winfield.

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.

Dallas Cowboys safety Roy Williams wore 31 before switching to 38.

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 code for an annoying person backstage at a concert
• 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)
• The number of the French department Haute-Garonne

Historical years

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

References

• Prime Curios! 31 from the Prime Pages