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 (2⁵ − 1)^[1] and the eighth Mersenne prime exponent, as well as the fourth primorial prime, and together with twenty-nine, a 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 also the 4th lucky prime^[2] and the 11th supersingular prime.^[3]

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

31 is the eighth happy number.^[7]

For the Steiner tree problem, 31 is the number of possible Steiner topologies for Steiner trees with 4 terminals.^[8]

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.^[9]

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

The cube root of 31 is the value of pi correct to four significant figures.

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 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_wE or ER_w can consist only of primes because every prime in the sequence will periodically divide further numbers.^{[citation needed]} Here, 31 divides every fifteenth number in 3_w1 (and 331 every 110th).

• The atomic number of gallium

AstronomyEdit

• 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

• Ice hockey goaltenders often wear the number 31.^{[citation needed]}

Thirty-one is also:

• The number of days in the months January, March, May, July, August, October and December
• The number of the date that Halloween and New Year's Eve are celebrated
• 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
• 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
• Number of letters in Ottoman alphabet
• The number of years approximately equal to 1 billion seconds

• Prime Curios! 31 from the Prime Pages