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 Super-prime numbers, also known as higher-order primes or prime-indexed primes (PIPs), are the subsequence of prime numbers that occupy prime-numbered positions within the sequence of all prime numbers. The subsequence begins :3, 5, 11, 17, 31, ...

and a self prime (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; two properties it shares with 13, which is its dual

emirp An emirp (''prime'' spelled backwards) is a prime number that results in a different prime when its decimal digits are reversed. This definition excludes the related palindromic primes. The term ''reversible prime'' is used to mean the same as ...

and

permutable prime A permutable prime, also known as anagrammatic prime, is a prime number which, in a given base, can have its digits' positions switched through any permutation and still be a prime number. H. E. Richert, who is supposedly the first to study th ...

. 31 is also a

primorial prime In mathematics, a primorial prime is a prime number of the form ''pn''# ± 1, where ''pn''# is the primorial of ''pn'' (i.e. the product of the first ''n'' primes). Primality tests show that : ''pn''# − 1 is prime for ''n ...

, 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 pr ...

, 29. 31 is the number of regular polygons with an odd number of sides that are known to be constructible with compass and straightedge, from combinations of known Fermat primes 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 17t ...

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 Daniel ...

, 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 calculation ...

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, e ...

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

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. T ...

496 __NOTOC__ Year 496 ( CDXCVI) was a leap year starting on Monday (link will display the full calendar) of the Julian calendar. In the Roman Empire, it was known as the Year of the Consulship of Paulus without colleague (or, less frequently, ye ...

, of the form 2^{(5 − 1)}(2⁵ − 1). 31 is a centered triangular number, the first prime centered pentagonal number and a centered decagonal number. For the Steiner tree problem, 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

110 110 may refer to: *110 (number), natural number *AD 110, a year *110 BC, a year *110 film, a cartridge-based film format used in still photography *110 (MBTA bus), Massachusetts Bay Transportation Authority bus route *110 (song), 2019 song by Capi ...

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 kind ...

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 Year 199 ( CXCIX) was a common year starting on Monday (link will display the full calendar) of the Julian calendar. At the time, it was sometimes known as year 952 ''Ab urbe condita''. The denomination 199 for this year has been used since the ...

× 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 A shape or figure is a graphics, graphical representation of an obje ...

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 const ...

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 figur ...

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 toge ...

, per

Moser's circle problem The number of and for first 6 terms of Moser's circle problem In geometry, the problem of dividing a circle into areas by means of an inscribed polygon with ''n'' sides in such a way as to ''maximise'' the number of areas created by the edges an ...

. 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 every ...

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

Astronomy

* Messier object 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 Asterism (astronomy), a perceived pattern or outline, typically representing an animal, mythological subject, or inanimate object. The origins of the e ...

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 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 NGC 31 is a spiral galaxy located in the constellation Phoenix. It was discovered on October 28, 1834 by the astronomer John Herschel. Its morphological type is SB(rs)cd, meaning that it is a late-type barred spiral galaxy A barred spiral gal ...

, 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 hock ...

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 board * The number of flavors of Baskin-Robbins

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 str ...

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

ISO ISO is the most common abbreviation for the International Organization for Standardization. ISO or Iso may also refer to: Business and finance * Iso (supermarket), a chain of Danish supermarkets incorporated into the SuperBest chain in 2007 * 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 Audiovisue ...

'' * In the title of Nick Hornby's book ''

31 Songs ''Songbook'' (published in the United Kingdom as ''31 Songs'') is a 2002 collection of 26 essays by English writer Nick Hornby about songs and (more often) the particular emotional resonance they carry for him. In the UK, Sony released a stand- ...

'' * 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's ...

* 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 (equ ...

), 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 of ...

and promulgated in the 20th century by Adriaan Fokker * Number of letters in

Macedonian alphabet The orthography of the Macedonian language includes an alphabet consisting of 31 letters ( mk, Македонска азбука, Makedonska azbuka), which is an adaptation of the Cyrillic script, as well as language-specific conventions of spelli ...

*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 {{Integers, zero Integers