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

In mathematics

31 is the 11th prime number. It is a superprime and a

self prime In number theory, a self number or Devlali number in a given number base b is a natural number that cannot be written as the sum of any other natural number n and the individual digits of n. 20 is a self number (in base 10), because no such combi ...

(after 3, 5, and 7), as no integer added up to its base 10 digits results in 31. It is a lucky prime and a

happy number In number theory, a happy number is a number which eventually reaches 1 when replaced by the sum of the square of each digit. For instance, 13 is a happy number because 1^2+3^2=10, and 1^2+0^2=1. On the other hand, 4 is not a happy number because ...

; two properties it shares with 13, which is its dual emirp and permutable prime. 31 is also a primorial prime, like its

twin prime A twin prime is a prime number that is either 2 less or 2 more than another prime number—for example, either member of the twin prime pair (41, 43). In other words, a twin prime is a prime that has a prime gap of two. Sometimes the term ''twin p ...

, 29. 31 is the number of

regular polygon In Euclidean geometry, a regular polygon is a polygon that is direct equiangular (all angles are equal in measure) and equilateral (all sides have the same length). Regular polygons may be either convex, star or skew. In the limit, a sequence ...

s with an odd number of sides that are known to be constructible with compass and straightedge, from combinations of known

Fermat prime In mathematics, a Fermat number, named after Pierre de Fermat, who first studied them, is a positive integer of the form :F_ = 2^ + 1, where ''n'' is a non-negative integer. The first few Fermat numbers are: : 3, 5, 17, 257, 65537, 429496 ...

s of the form 2^{2^''n''} + 1. 31 is the third

Mersenne prime In mathematics, a Mersenne prime is a prime number that is one less than a power of two. That is, it is a prime number of the form for some integer . They are named after Marin Mersenne, a French Minim friar, who studied them in the early 17 ...

of the form 2^''n'' − 1. It is also the eighth Mersenne prime exponent, specifically for the number

2,147,483,647 The number 2,147,483,647 is the eighth Mersenne prime, equal to 231 − 1. It is one of only four known double Mersenne primes. The primality of this number was proven by Leonhard Euler, who reported the proof in a letter to Dani ...

, which is the maximum positive value for a

32-bit In computer architecture, 32-bit computing refers to computer systems with a processor, memory, and other major system components that operate on data in 32- bit units. Compared to smaller bit widths, 32-bit computers can perform large calculati ...

signed binary integer in

computing Computing is any goal-oriented activity requiring, benefiting from, or creating computing machinery. It includes the study and experimentation of algorithmic processes, and development of both hardware and software. Computing has scientific, ...

. After 3, it is the second Mersenne prime not to be a

double Mersenne prime In mathematics, a double Mersenne number is a Mersenne number of the form :M_ = 2^-1 where ''p'' is prime. Examples The first four terms of the sequence of double Mersenne numbers areChris Caldwell''Mersenne Primes: History, Theorems and ...

. 127, which is the 31st prime number, is a double Mersenne prime. The 31st

triangular number A triangular number or triangle number counts objects arranged in an equilateral triangle. Triangular numbers are a type of figurate number, other examples being square numbers and cube numbers. The th triangular number is the number of dots i ...

is the

perfect number In number theory, a perfect number is a positive integer that is equal to the sum of its positive divisors, excluding the number itself. For instance, 6 has divisors 1, 2 and 3 (excluding itself), and 1 + 2 + 3 = 6, so 6 is a perfect number. ...

496, of the form 2^{(5 − 1)}(2⁵ − 1). 31 is a

centered triangular number A centered (or centred) triangular number is a centered figurate number that represents an equilateral triangle with a dot in the center and all its other dots surrounding the center in successive equilateral triangular layers. The followin ...

, the first prime

centered pentagonal number A centered pentagonal number is a centered figurate number that represents a pentagon with a dot in the center and all other dots surrounding the center in successive pentagonal layers. The centered pentagonal number for ''n'' is given by th ...

and a

centered decagonal number A centered decagonal number is a centered figurate number that represents a decagon with a dot in the center and all other dots surrounding the center dot in successive decagonal layers. The centered decagonal number for ''n'' is given by th ...

. For the

Steiner tree problem In combinatorial mathematics, the Steiner tree problem, or minimum Steiner tree problem, named after Jakob Steiner, is an umbrella term for a class of problems in combinatorial optimization. While Steiner tree problems may be formulated in a ...

, 31 is the number of possible Steiner topologies for Steiner trees with 4 terminals. At 31, the Mertens function sets a new low of −4, a value which is not subceded until 110. Centered pentagonal number 31

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, , , , and 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 Composite or compositing may refer to: Materials * Composite material, a material that is made from several different substances ** Metal matrix composite, composed of metal and other parts ** Cermet, a composite of ceramic and metallic materials ...

; their

factorisation In mathematics, factorization (or factorisation, see English spelling differences) or factoring consists of writing a number or another mathematical object as a product of several ''factors'', usually smaller or simpler objects of the same kin ...

s are: * = 17 × * = 673 × * = 307 × * = 19 × 83 × * = 523 × 3049 × * = 607 × 1511 × 1997 × * =

181 Year 181 ( CLXXXI) was a common year starting on Sunday (link will display the full calendar) of the Julian calendar. At the time, it was known as the Year of the Consulship of Aurelius and Burrus (or, less frequently, year 934 ''Ab urbe condit ...

× * = 199 × and * = 31 × 1499 × . 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). 31 is the 11th and final consecutive supersingular prime. After 31, the only supersingular primes are 41, 47, 59, and 71. 31 is the maximum number of

area Area is the quantity that expresses the extent of a region on the plane or on a curved surface. The area of a plane region or ''plane area'' refers to the area of a shape or planar lamina, while '' surface area'' refers to the area of an op ...

s inside a

circle A circle is a shape consisting of all points in a plane that are at a given distance from a given point, the centre. Equivalently, it is the curve traced out by a point that moves in a plane so that its distance from a given point is cons ...

created from the edges and diagonals of an

inscribed {{unreferenced, date=August 2012 An inscribed triangle of a circle In geometry, an inscribed planar shape or solid is one that is enclosed by and "fits snugly" inside another geometric shape or solid. To say that "figure F is inscribed in figu ...

six-sided

polygon In geometry, a polygon () is a plane figure that is described by a finite number of straight line segments connected to form a closed '' polygonal chain'' (or ''polygonal circuit''). The bounded plane region, the bounding circuit, or the two ...

, per Moser's circle problem. It is also equal to the sum of the maximum number of areas generated by the first five ''n''-sided polygons: 1, 2, 4, 8, 16, and as such, 31 is the first member that diverges from twice the value of its previous member in the sequence, by 1.

In science

* The

atomic number The atomic number or nuclear charge number (symbol ''Z'') of a chemical element is the charge number of an atomic nucleus. For ordinary nuclei, this is equal to the proton number (''n''p) or the number of protons found in the nucleus of ever ...

gallium Gallium is a chemical element with the Symbol (chemistry), symbol Ga and atomic number 31. Discovered by France, French chemist Paul-Émile Lecoq de Boisbaudran in 1875, Gallium is in boron group, group 13 of the periodic table and is similar to ...

Astronomy

Messier object The Messier objects are a set of 110 astronomical objects catalogued by the French astronomer Charles Messier in his ''Catalogue des Nébuleuses et des Amas d'Étoiles'' (''Catalogue of Nebulae and Star Clusters''). Because Messier was only in ...

M31, a magnitude 4.5

galaxy A galaxy is a system of stars, stellar remnants, interstellar gas, dust, dark matter, bound together by gravity. The word is derived from the Greek ' (), literally 'milky', a reference to the Milky Way galaxy that contains the Solar System ...

in the

constellation A constellation is an area on the celestial sphere in which a group of visible stars forms a perceived pattern or outline, typically representing an animal, mythological subject, or inanimate object. The origins of the earliest constellation ...

Andromeda. It is also known as the

Andromeda Galaxy The Andromeda Galaxy (IPA: ), also known as Messier 31, M31, or NGC 224 and originally the Andromeda Nebula, is a barred spiral galaxy with the diameter of about approximately from Earth and the nearest large galaxy to the Milky Way. The gal ...

, and is readily visible to the naked eye in a modestly dark sky. * The

New General Catalogue The ''New General Catalogue of Nebulae and Clusters of Stars'' (abbreviated NGC) is an astronomical catalogue of deep-sky objects compiled by John Louis Emil Dreyer in 1888. The NGC contains 7,840 objects, including galaxies, star clusters and ...

br>object
NGC 31, a

spiral galaxy Spiral galaxies form a class of galaxy originally described by Edwin Hubble in his 1936 work ''The Realm of the Nebulae''Phoenix

In sports

Ice hockey Ice hockey (or simply hockey) is a team sport played on ice skates, usually on an ice skating rink with lines and markings specific to the sport. It belongs to a family of sports called hockey. In ice hockey, two opposing teams use ice ...

goaltenders often wear the number 31.

In other fields

Thirty-one is also: * The number of days in each of 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 31 (thirty-one) is the natural number following 30 and preceding 32. It is a prime number. In mathematics 31 is the 11th prime number. It is a superprime and a self prime (after 3, 5, and 7), as no integer added up to its base 10 digits ...

, a card game *The number of kings defeated by the incoming Israelite settlers in

Canaan Canaan (; Phoenician: 𐤊𐤍𐤏𐤍 – ; he, כְּנַעַן – , in pausa – ; grc-bib, Χανααν – ;The current scholarly edition of the Greek Old Testament spells the word without any accents, cf. Septuaginta : id est Vetus T ...

according to : "all the kings, one and thirty" ( Wycliffe Bible translation) *A type of game played on a

backgammon Backgammon is a two-player board game played with counters and dice on tables boards. It is the most widespread Western member of the large family of tables games, whose ancestors date back nearly 5,000 years to the regions of Mesopotamia and Pe ...

board * The number of flavors of

Baskin-Robbins Baskin-Robbins is an American multinational chain of ice cream and cake speciality shops owned by Inspire Brands. Based in Canton, Massachusetts, Baskin-Robbins was founded in 1945 by Burt Baskin (1913–1967) and Irv Robbins (1917–2008) in ...

ice cream Ice cream is a sweetened frozen food typically eaten as a snack or dessert. It may be made from milk or cream and is flavoured with a sweetener, either sugar or an alternative, and a spice, such as cocoa or vanilla, or with fruit such as ...

; the shops are called '' 31 Ice Cream'' in Japan *

ISO 31 ISO 31 ( Quantities and units, International Organization for Standardization, 1992) is a superseded international standard concerning physical quantities, units of measurement, their interrrelationships and their presentation. It was revised and ...

is the ISO's standard for quantities and units * In the title of the anime ''

Ulysses 31 (french: link=no, Ulysse 31) is an anime series (1981) that updates the Greek mythology of Odysseus (known as "Ulixes" or "Ulysses" in Latin) to the 31st century. The show comprises 26 half-hour episodes as a co-production between DIC Audio ...

'' * 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 Haute-Garonne (; oc, Nauta Garona, ; en, Upper Garonne) is a department in the Occitanie region of Southwestern France. Named after the river Garonne, which flows through the department. Its prefecture and main city is Toulouse, the country' ...

* In music, 31-tone equal temperament is a historically significant tuning system (

31 equal temperament In music, 31 equal temperament, 31-ET, which can also be abbreviated 31-TET (31 tone ET) or 31-EDO (equal division of the octave), also known as tricesimoprimal, is the tempered scale derived by dividing the octave into 31 equal-sized steps (equa ...

), first theorized by

Christiaan Huygens Christiaan Huygens, Lord of Zeelhem, ( , , ; also spelled Huyghens; la, Hugenius; 14 April 1629 – 8 July 1695) was a Dutch mathematician, physicist, engineer, astronomer, and inventor, who is regarded as one of the greatest scientists o ...

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

References

External links

Prime Curios! 31
from the

Prime Pages The PrimePages is a website about prime numbers maintained by Chris Caldwell at the University of Tennessee at Martin. The site maintains the list of the "5,000 largest known primes", selected smaller primes of special forms, and many "top twenty" ...

{{Integers, zero Integers