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 results in 31. It is a
lucky prime
In number theory, a lucky number is a natural number in a set which is generated by a certain "sieve". This sieve is similar to the Sieve of Eratosthenes that generates the primes, but it eliminates numbers based on their position in the remaini ...
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
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 polygon
In Euclidean geometry, a regular polygon is a polygon that is Equiangular polygon, direct equiangular (all angles are equal in measure) and Equilateral polygon, equilateral (all sides have the same length). Regular polygons may be either convex p ...
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, 4294967 ...
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 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
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 Li ...
.
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, of the form 2
(5 − 1)(2
5 − 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 following ...
, 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 the ...
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 the ...
.
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 nu ...
, 31 is the number of possible Steiner topologies for Steiner trees with 4 terminals.
At 31, the
Mertens function
In number theory, the Mertens function is defined for all positive integers ''n'' as
: M(n) = \sum_^n \mu(k),
where \mu(k) is the Möbius function. The function is named in honour of Franz Mertens. This definition can be extended to positive r ...
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 ...
.
![Centered pentagonal number 31](https://upload.wikimedia.org/wikipedia/commons/9/94/Centered_pentagonal_number_31.svg)
31 is a
repdigit
In recreational mathematics, a repdigit or sometimes monodigit is a natural number composed of repeated instances of the same digit in a positional number system (often implicitly decimal). The word is a portmanteau of repeated and digit.
Example ...
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
factorisations are:
* =
17 ×
* = 673 ×
* = 307 ×
* =
19 ×
83 ×
* = 523 × 3049 ×
* = 607 × 1511 × 1997 ×
* =
181 ×
* =
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
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.
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 ...
of
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
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 int ...
M31, a
magnitude
Magnitude may refer to:
Mathematics
*Euclidean vector, a quantity defined by both its magnitude and its direction
*Magnitude (mathematics), the relative size of an object
*Norm (mathematics), a term for the size or length of a vector
*Order of ...
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
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 gala ...
, 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>
objectNGC 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
Phoenix most often refers to:
* Phoenix (mythology), a legendary bird from ancient Greek folklore
* Phoenix, Arizona, a city in the United States
Phoenix may also refer to:
Mythology
Greek mythological figures
* Phoenix (son of Amyntor), a ...]
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
Wycliffe's Bible is the name now given to a group of Bible translations into Middle English that were made under the direction of English theologian John Wycliffe. They appeared over a period from approximately 1382 to 1395. These Bible translati ...
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 str ...
; 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
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
Adriaan Daniël Fokker (; 17 August 1887 – 24 September 1972) was a Dutch physicist. He worked in the fields of special relativity and statistical mechanics. He was the inventor of the Fokker organ, a 31-tone equal-tempered (31-TET) organ. ...
* 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 Ottoman Turkish alphabet ( ota, الفبا, ') is a version of the Arabic script used to write Ottoman Turkish
Ottoman Turkish ( ota, لِسانِ عُثمانى, Lisân-ı Osmânî, ; tr, Osmanlı Türkçesi) was the standardized regi ...
* The number of years approximately equal to 1 billion seconds
References
External links
Prime Curios! 31from 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