Category: Number theory

Inter-universal Teichmüller theory
Inter-universal Teichmüller theory (abbreviated as IUT or IUTT) is the name given by mathematician Shinichi Mochizuki to a theory he developed in the 2000s, following his earlier work in arithmetic ge
Eichler order
In mathematics, an Eichler order, named after Martin Eichler, is an order of a quaternion algebra that is the intersection of two maximal orders.
Perfect power
In mathematics, a perfect power is a natural number that is a product of equal natural factors, or, in other words, an integer that can be expressed as a square or a higher integer power of another in
Prouhet–Thue–Morse constant
In mathematics, the Prouhet–Thue–Morse constant, named for , Axel Thue, and Marston Morse, is the number—denoted by τ—whose binary expansion 0.01101001100101101001011001101001... is given by the Thue–
Automorphic number
In mathematics, an automorphic number (sometimes referred to as a circular number) is a natural number in a given number base whose square "ends" in the same digits as the number itself.
Bhargava cube
In mathematics, in number theory, a Bhargava cube (also called Bhargava's cube) is a configuration consisting of eight integers placed at the eight corners of a cube. This configuration was extensivel
Dedekind sum
In mathematics, Dedekind sums are certain sums of products of a sawtooth function, and are given by a function D of three integer variables. Dedekind introduced them to express the functional equation
Eventually (mathematics)
In the mathematical areas of number theory and analysis, an infinite sequence or a function is said to eventually have a certain property, if it doesn't have the said property across all its ordered i
Sociable number
In mathematics, sociable numbers are numbers whose aliquot sums form a periodic sequence. They are generalizations of the concepts of amicable numbers and perfect numbers. The first two sociable seque
Erdős–Tenenbaum–Ford constant
The Erdős–Tenenbaum–Ford constant is a mathematical constant that appears in number theory. Named after mathematicians Paul Erdős, Gérald Tenenbaum, and Kevin Ford, it is defined as where is the natur
Seventh power
In arithmetic and algebra the seventh power of a number n is the result of multiplying seven instances of n together. So: n7 = n × n × n × n × n × n × n. Seventh powers are also formed by multiplying
Schoof–Elkies–Atkin algorithm
The Schoof–Elkies–Atkin algorithm (SEA) is an algorithm used for finding the order of or calculating the number of points on an elliptic curve over a finite field. Its primary application is in ellipt
Number Theory Foundation
The Number Theory Foundation (NTF) is a non-profit organization based in the United States which supports research and conferences in the field of number theory, with a particular focus on computation
Prosolvable group
In mathematics, more precisely in algebra, a prosolvable group (less common: prosoluble group) is a group that is isomorphic to the inverse limit of an inverse system of solvable groups. Equivalently,
Grothendieck–Teichmüller group
In mathematics, the Grothendieck–Teichmüller group GT is a group closely related to (and possibly equal to) the absolute Galois group of the rational numbers. It was introduced by Vladimir Drinfeld an
Dyadic rational
In mathematics, a dyadic rational or binary rational is a number that can be expressed as a fraction whose denominator is a power of two. For example, 1/2, 3/2, and 3/8 are dyadic rationals, but 1/3 i
Erdős arcsine law
In number theory, the Erdős arcsine law, named after Paul Erdős in 1969, states that the prime divisors of a number have a distribution related to the arcsine distribution. Specifically, say that the
Golomb ruler
In mathematics, a Golomb ruler is a set of marks at integer positions along a ruler such that no two pairs of marks are the same distance apart. The number of marks on the ruler is its order, and the
Proofs of Fermat's little theorem
This article collects together a variety of proofs of Fermat's little theorem, which states that for every prime number p and every integer a (see modular arithmetic).
Tate twist
In number theory and algebraic geometry, the Tate twist, named after John Tate, is an operation on Galois modules. For example, if K is a field, GK is its absolute Galois group, and ρ : GK → AutQp(V)
Sixth power
In arithmetic and algebra the sixth power of a number n is the result of multiplying six instances of n together. So: n6 = n × n × n × n × n × n. Sixth powers can be formed by multiplying a number by
Unitary divisor
In mathematics, a natural number a is a unitary divisor (or Hall divisor) of a number b if a is a divisor of b and if a and are coprime, having no common factor other than 1. Thus, 5 is a unitary divi
An Introduction to the Theory of Numbers
An Introduction to the Theory of Numbers is a classic textbook in the field of number theory, by G. H. Hardy and E. M. Wright. The book grew out of a series of lectures by Hardy and Wright and was fir
Hall–Higman theorem
In mathematical group theory, the Hall–Higman theorem, due to Philip Hall and Graham Higman , describes the possibilities for the minimal polynomial of an element of prime power order for a representa
Euler function
In mathematics, the Euler function is given by Named after Leonhard Euler, it is a model example of a q-series and provides the prototypical example of a relation between combinatorics and complex ana
In mathematics, E-functions are a type of power series that satisfy particular arithmetic conditions on the coefficients. They are of interest in transcendental number theory, and are more special tha
Euler's totient function
In number theory, Euler's totient function counts the positive integers up to a given integer n that are relatively prime to n. It is written using the Greek letter phi as or , and may also be called
Bernoulli number
In mathematics, the Bernoulli numbers Bn are a sequence of rational numbers which occur frequently in analysis. The Bernoulli numbers appear in (and can be defined by) the Taylor series expansions of
Ideal number
In number theory an ideal number is an algebraic integer which represents an ideal in the ring of integers of a number field; the idea was developed by Ernst Kummer, and led to Richard Dedekind's defi
Prime signature
In mathematics, the prime signature of a number is the multiset of (nonzero) exponents of its prime factorization. The prime signature of a number having prime factorization is the multiset . For exam
Fermat quotient
In number theory, the Fermat quotient of an integer a with respect to an odd prime p is defined as or . This article is about the former; for the latter see p-derivation. The quotient is named after P
Numerical semigroup
In mathematics, a numerical semigroup is a special kind of a semigroup. Its underlying set is the set of all nonnegative integers except a finite number and the binary operation is the operation of ad
Signed-digit representation
In mathematical notation for numbers, a signed-digit representation is a positional numeral system with a set of signed digits used to encode the integers. Signed-digit representation can be used to a
Distribution (number theory)
In algebra and number theory, a distribution is a function on a system of finite sets into an abelian group which is analogous to an integral: it is thus the algebraic analogue of a distribution in th
Aurifeuillean factorization
In number theory, an aurifeuillean factorization, named after Léon-François-Antoine Aurifeuille, is a special type of algebraic factorization that comes from non-trivial factorizations of cyclotomic p
Amicable triple
In mathematics, an amicable triple is a set of three different numbers so related that the restricted sum of the divisors of each is equal to the sum of other two numbers. In another equivalent charac
Deformation ring
In mathematics, a deformation ring is a ring that controls liftings of a representation of a Galois group from a finite field to a local field. In particular for any such lifting problem there is ofte
Quadratic reciprocity
In number theory, the law of quadratic reciprocity is a theorem about modular arithmetic that gives conditions for the solvability of quadratic equations modulo prime numbers. Due to its subtlety, it
3x + 1 semigroup
In algebra, the 3x + 1 semigroup is a special subsemigroup of the multiplicative semigroup of all positive rational numbers. The elements of a generating set of this semigroup are related to the seque
Eighth power
In arithmetic and algebra the eighth power of a number n is the result of multiplying eight instances of n together. So: n8 = n × n × n × n × n × n × n × n. Eighth powers are also formed by multiplyin
Journal of Number Theory
The Journal of Number Theory (JNT) is a bimonthly peer-reviewed scientific journal covering all aspects of number theory. The journal was established in 1969 by R.P. Bambah, P. Roquette, A. Ross, A. W
Ribenboim Prize
The Ribenboim Prize, named in honour of Paulo Ribenboim, is awarded by the Canadian Number Theory Association for distinguished research in number theory by a mathematician who is Canadian or has clos
Multiplicative partition
In number theory, a multiplicative partition or unordered factorization of an integer n is a way of writing n as a product of integers greater than 1, treating two products as equivalent if they diffe
Arithmetic derivative
In number theory, the Lagarias arithmetic derivative or number derivative is a function defined for integers, based on prime factorization, by analogy with the product rule for the derivative of a fun
Büchi's problem
In number theory, Büchi's problem, also known as the n squares' problem, is an open problem named after the Swiss mathematician Julius Richard Büchi. It asks whether there is a positive integer M such
List of number theory topics
This is a list of number theory topics, by Wikipedia page. See also: * List of recreational number theory topics * Topics in cryptography
Irrationality sequence
In mathematics, a sequence of positive integers an is called an irrationality sequence if it has the property that for every sequence xn of positive integers, the sum of the series exists (that is, it
Natural density
In number theory, natural density (also referred to as asymptotic density or arithmetic density) is one method to measure how "large" a subset of the set of natural numbers is. It relies chiefly on th
Prime geodesic
In mathematics, a prime geodesic on a hyperbolic surface is a primitive closed geodesic, i.e. a geodesic which is a closed curve that traces out its image exactly once. Such geodesics are called prime
Birch and Swinnerton-Dyer conjecture
In mathematics, the Birch and Swinnerton-Dyer conjecture (often called the Birch–Swinnerton-Dyer conjecture) describes the set of rational solutions to equations defining an elliptic curve. It is an o
Generalized taxicab number
In mathematics, the generalized taxicab number Taxicab(k, j, n) is the smallest number — if it exists — that can be expressed as the sum of j kth positive powers in n different ways. For k = 3 and j =
Squared triangular number
In number theory, the sum of the first n cubes is the square of the nth triangular number. That is, The same equation may be written more compactly using the mathematical notation for summation: This
Bernoulli polynomials of the second kind
The Bernoulli polynomials of the second kind ψn(x), also known as the Fontana-Bessel polynomials, are the polynomials defined by the following generating function: The first five polynomials are: Some
Gauss's diary
Gauss's diary was a record of the mathematical discoveries of German mathematician Carl Friedrich Gauss from 1796 to 1814. It was rediscovered in 1897 and published by , and reprinted in volume X1 of
Barnes G-function
In mathematics, the Barnes G-function G(z) is a function that is an extension of superfactorials to the complex numbers. It is related to the gamma function, the K-function and the Glaisher–Kinkelin c
Vieta jumping
In number theory, Vieta jumping, also known as root flipping, is a proof technique. It is most often used for problems in which a relation between two positive integers is given, along with a statemen
Wild number
Originally, wild numbers are the numbers supposed to belong to a fictional sequence of numbers imagined to exist in the mathematical world of the mathematical fiction The Wild Numbers authored by Phil
P-adic number
In mathematics, the p-adic number system for any prime number p extends the ordinary arithmetic of the rational numbers in a different way from the extension of the rational number system to the real
Local analysis
In mathematics, the term local analysis has at least two meanings, both derived from the idea of looking at a problem relative to each prime number p first, and then later trying to integrate the info
Cunningham number
In mathematics, specifically in number theory, a Cunningham number is a certain kind of integer named after English mathematician A. J. C. Cunningham.
Hecke character
In number theory, a Hecke character is a generalisation of a Dirichlet character, introduced by Erich Hecke to construct a class ofL-functions larger than Dirichlet L-functions, and a natural setting
Divisor function
In mathematics, and specifically in number theory, a divisor function is an arithmetic function related to the divisors of an integer. When referred to as the divisor function, it counts the number of
Integer-valued polynomial
In mathematics, an integer-valued polynomial (also known as a numerical polynomial) is a polynomial whose value is an integer for every integer n. Every polynomial with integer coefficients is integer
Schoof's algorithm
Schoof's algorithm is an efficient algorithm to count points on elliptic curves over finite fields. The algorithm has applications in elliptic curve cryptography where it is important to know the numb
Square-free integer
In mathematics, a square-free integer (or squarefree integer) is an integer which is divisible by no square number other than 1. That is, its prime factorization has exactly one factor for each prime
Primon gas
In mathematical physics, the primon gas or free Riemann gas is a toy model illustrating in a simple way some correspondences between number theory and ideas in quantum field theory and dynamical syste
Hundred Fowls Problem
The Hundred Fowls Problem is a problem first discussed in the fifth century CE Chinese mathematics text Zhang Qiujian suanjing (The Mathematical Classic of Zhang Qiujian), a book of mathematical probl
Hydra game
In mathematics, specifically in graph theory and number theory, a hydra game is a single-player iterative mathematical game played on a mathematical tree called a hydra where, usually, the goal is to
Artin conductor
In mathematics, the Artin conductor is a number or ideal associated to a character of a Galois group of a local or global field, introduced by Emil Artin as an expression appearing in the functional e
Formulas for generating Pythagorean triples
Besides Euclid's formula, many other formulas for generating Pythagorean triples have been developed.
Granville number
In mathematics, specifically number theory, Granville numbers, also known as -perfect numbers, are an extension of the perfect numbers.
Frey curve
In mathematics, a Frey curve or Frey–Hellegouarch curve is the elliptic curve associated with a (hypothetical) solution of Fermat's equation The curve is named after Gerhard Frey.
Multiplicative independence
In number theory, two positive integers a and b are said to be multiplicatively independent if their only common integer power is 1. That is, for integers n and m, implies . Two integers which are not
Upper half-plane
In mathematics, the upper half-plane, is the set of points (x, y) in the Cartesian plane with y > 0.
Elliptic divisibility sequence
In mathematics, an elliptic divisibility sequence (EDS) is a sequence of integers satisfying a nonlinear recursion relation arising from division polynomials on elliptic curves. EDS were first defined
Crank conjecture
In mathematics, the crank conjecture was a conjecture about the existence of the crank of a partition that separates partitions of a number congruent to 6 mod 11 into 11 equal classes. The conjecture
Weyl's inequality (number theory)
In number theory, Weyl's inequality, named for Hermann Weyl, states that if M, N, a and q are integers, with a and q coprime, q > 0, and f is a real polynomial of degree k whose leading coefficient c
Sums of powers
In mathematics and statistics, sums of powers occur in a number of contexts: * Sums of squares arise in many contexts. For example, in geometry, the Pythagorean theorem involves the sum of two square
Greedy algorithm for Egyptian fractions
In mathematics, the greedy algorithm for Egyptian fractions is a greedy algorithm, first described by Fibonacci, for transforming rational numbers into Egyptian fractions. An Egyptian fraction is a re
Vorlesungen über Zahlentheorie
Vorlesungen über Zahlentheorie (German for Lectures on Number Theory) is the name of several different textbooks of number theory. The best known was written by Peter Gustav Lejeune Dirichlet and Rich
Schnorr group
A Schnorr group, proposed by Claus P. Schnorr, is a large prime-order subgroup of , the multiplicative group of integers modulo for some prime . To generate such a group, generate , , such that with ,
Binomial number
In mathematics, specifically in number theory, a binomial number is an integer which can be obtained by evaluating a homogeneous polynomial containing two terms. It is a generalization of a Cunningham
Divisor sum identities
The purpose of this page is to catalog new, interesting, and useful identities related to number-theoretic divisor sums, i.e., sums of an arithmetic function over the divisors of a natural number , or
Cunningham Project
The Cunningham Project is a collaborative effort started in 1925 to factor numbers of the form bn ± 1 for b = 2, 3, 5, 6, 7, 10, 11, 12 and large n. The project is named after Allan Joseph Champneys C
Auxiliary function
In mathematics, auxiliary functions are an important construction in transcendental number theory. They are functions that appear in most proofs in this area of mathematics and that have specific, des
Arithmetic Fuchsian group
Arithmetic Fuchsian groups are a special class of Fuchsian groups constructed using orders in quaternion algebras. They are particular instances of arithmetic groups. The prototypical example of an ar
Schwartz–Bruhat function
In mathematics, a Schwartz–Bruhat function, named after Laurent Schwartz and François Bruhat, is a complex valued function on a locally compact abelian group, such as the adeles, that generalizes a Sc
Abc conjecture
The abc conjecture (also known as the Oesterlé–Masser conjecture) is a conjecture in number theory that arose out of a discussion of Joseph Oesterlé and David Masser in 1985. It is stated in terms of
Digit sum
In mathematics, the digit sum of a natural number in a given number base is the sum of all its digits. For example, the digit sum of the decimal number would be .
Hyperharmonic number
In mathematics, the n-th hyperharmonic number of order r, denoted by , is recursively defined by the relations: and In particular, is the n-th harmonic number. The hyperharmonic numbers were discussed
List of types of functions
Functions can be identified according to the properties they have. These properties describe the functions' behaviour under certain conditions. A parabola is a specific type of function.
Cube (algebra)
In arithmetic and algebra, the cube of a number n is its third power, that is, the result of multiplying three instances of n together.The cube of a number or any other mathematical expression is deno
Fontaine–Mazur conjecture
In mathematics, the Fontaine–Mazur conjectures are some conjectures introduced by Fontaine and Mazur about when p-adic representations of Galois groups of number fields can be constructed from represe
List of mathematical functions
In mathematics, some functions or groups of functions are important enough to deserve their own names. This is a listing of articles which explain some of these functions in more detail. There is a la
Normal number
In mathematics, a real number is said to be simply normal in an integer base b if its infinite sequence of digits is distributed uniformly in the sense that each of the b digit values has the same nat
Quadratic irrational number
In mathematics, a quadratic irrational number (also known as a quadratic irrational, a quadratic irrationality or quadratic surd) is an irrational number that is the solution to some quadratic equatio
Skew binary number system
The skew binary number system is a non-standard positional numeral system in which the nth digit contributes a value of times the digit (digits are indexed from 0) instead of times as they do in binar
Extended natural numbers
In mathematics, the extended natural numbers is a set which contains the values and (infinity). That is, it is result of adding a maximum element to the natural numbers. Addition and multiplication wo
P-adic distribution
In mathematics, a p-adic distribution is an analogue of ordinary distributions (i.e. generalized functions) that takes values in a ring of p-adic numbers.
Natural number
In mathematics, the natural numbers are those numbers used for counting (as in "there are six coins on the table") and ordering (as in "this is the third largest city in the country"). Numbers used fo
Amenable number
An amenable number is a positive integer for which there exists a multiset of as many integers as the original number that both add up to the original number and when multiplied together give the orig
Hurwitz class number
In mathematics, the Hurwitz class number H(N), introduced by Adolf Hurwitz, is a modification of the class number of positive definite binary quadratic forms of discriminant –N, where forms are weight
Integer square root
In number theory, the integer square root (isqrt) of a non-negative integer n is the non-negative integer m which is the greatest integer less than or equal to the square root of n, For example,
Niven's constant
In number theory, Niven's constant, named after Ivan Niven, is the largest exponent appearing in the prime factorization of any natural number n "on average". More precisely, if we define H(1) = 1 and
Strict differentiability
In mathematics, strict differentiability is a modification of the usual notion of differentiability of functions that is particularly suited to p-adic analysis. In short, the definition is made more r
Ramanujan's sum
In number theory, Ramanujan's sum, usually denoted cq(n), is a function of two positive integer variables q and n defined by the formula where (a, q) = 1 means that a only takes on values coprime to q
Taxicab number
In mathematics, the nth taxicab number, typically denoted Ta(n) or Taxicab(n), also called the nth Hardy–Ramanujan number, is defined as the smallest integer that can be expressed as a sum of two posi
Poussin proof
In number theory, the Poussin proof is the proof of an identity related to the fractional part of a ratio. In 1838, Peter Gustav Lejeune Dirichlet proved an approximate formula for the average number
Digital root
The digital root (also repeated digital sum) of a natural number in a given radix is the (single digit) value obtained by an iterative process of summing digits, on each iteration using the result fro
Gan–Gross–Prasad conjecture
In mathematics, the Gan–Gross–Prasad conjecture is a restriction problem in the representation theory of real or p-adic Lie groups posed by Gan Wee Teck, Benedict Gross, and Dipendra Prasad. The probl
Cyclic number (group theory)
A cyclic number is a natural number n such that n and φ(n) are coprime. Here φ is Euler's totient function. An equivalent definition is that a number n is cyclic if and only if any group of order n is
Ducci sequence
A Ducci sequence is a sequence of n-tuples of integers, sometimes known as "the Diffy game", because it is based on sequences. Given an n-tuple of integers , the next n-tuple in the sequence is formed
Cyclotomic polynomial
In mathematics, the nth cyclotomic polynomial, for any positive integer n, is the unique irreducible polynomial with integer coefficients that is a divisor of and is not a divisor of for any k < n. It
Eisenstein–Kronecker number
In mathematics, Eisenstein–Kronecker numbers are an analogue for imaginary quadratic fields of generalized Bernoulli numbers. They are defined in terms of classical Eisenstein–Kronecker series, which
Fermat Prize
The Fermat prize of mathematical research biennially rewards research works in fields where the contributions of Pierre de Fermat have been decisive: * Statements of variational principles * Foundat
Least common multiple
In arithmetic and number theory, the least common multiple, lowest common multiple, or smallest common multiple of two integers a and b, usually denoted by lcm(a, b), is the smallest positive integer
Redheffer matrix
In mathematics, a Redheffer matrix, often denoted as studied by , is a square (0,1) matrix whose entries aij are 1 if i divides j or if j = 1; otherwise, aij = 0. It is useful in some contexts to expr
Hilbert's inequality
In analysis, a branch of mathematics, Hilbert's inequality states that for any sequence u1,u2,... of complex numbers. It was first demonstrated by David Hilbert with the constant 2π instead of π; the
Ideal lattice
In discrete mathematics, ideal lattices are a special class of lattices and a generalization of . Ideal lattices naturally occur in many parts of number theory, but also in other areas. In particular,
Znám's problem
In number theory, Znám's problem asks which sets of k integers have the property that each integer in the set is a proper divisor of the product of the other integers in the set, plus 1. Znám's proble
Mirimanoff's congruence
In number theory, a branch of mathematics, a Mirimanoff's congruence is one of a collection of expressions in modular arithmetic which, if they hold, entail the truth of Fermat's Last Theorem. Since t
Short integer solution problem
Short integer solution (SIS) and ring-SIS problems are two average-case problems that are used in lattice-based cryptography constructions. Lattice-based cryptography began in 1996 from a seminal work
Composition (combinatorics)
In mathematics, a composition of an integer n is a way of writing n as the sum of a sequence of (strictly) positive integers. Two sequences that differ in the order of their terms define different com
Dirichlet series inversion
In analytic number theory, a Dirichlet series, or Dirichlet generating function (DGF), of a sequence is a common way of understanding and summing arithmetic functions in a meaningful way. A little kno
Brewer sum
In mathematics, Brewer sums are finite character sum introduced by Brewer related to Jacobsthal sums.
Egyptian fraction
An Egyptian fraction is a finite sum of distinct unit fractions, such as That is, each fraction in the expression has a numerator equal to 1 and a denominator that is a positive integer, and all the d
Kubilius model
In mathematics, the Kubilius model relies on a clarification and extension of a finite probability space on which the behaviour of additive arithmetic functions can be modeled by sum of independent ra
Multiplicative digital root
In number theory, the multiplicative digital root of a natural number in a given number base is found by multiplying the digits of together, then repeating this operation until only a single-digit rem
Disquisitiones Arithmeticae
The Disquisitiones Arithmeticae (Latin for "Arithmetical Investigations") is a textbook of number theory written in Latin by Carl Friedrich Gauss in 1798 when Gauss was 21 and first published in 1801
Néron–Tate height
In number theory, the Néron–Tate height (or canonical height) is a quadratic form on the Mordell–Weil group of rational points of an abelian variety defined over a global field. It is named after Andr
Sparse ruler
A sparse ruler is a ruler in which some of the distance marks may be missing. More abstractly, a sparse ruler of length with marks is a sequence of integers where . The marks and correspond to the end
Timeline of number theory
A timeline of number theory.
Supernatural number
In mathematics, the supernatural numbers, sometimes called generalized natural numbers or Steinitz numbers, are a generalization of the natural numbers. They were used by Ernst Steinitz in 1910 as a p
Q-Pochhammer symbol
In mathematical area of combinatorics, the q-Pochhammer symbol, also called the q-shifted factorial, is the product with It is a q-analog of the Pochhammer symbol , in the sense thatThe q-Pochhammer s
Néron model
In algebraic geometry, the Néron model (or Néron minimal model, or minimal model)for an abelian variety AK defined over the field of fractions K of a Dedekind domain R is the "push-forward" of AK from
Tate–Shafarevich group
In arithmetic geometry, the Tate–Shafarevich group Ш(A/K) of an abelian variety A (or more generally a group scheme) defined over a number field K consists of the elements of the Weil–Châtelet group W
History of the Theory of Numbers
History of the Theory of Numbers is a three-volume work by L. E. Dickson summarizing work in number theory up to about 1920. The style is unusual in that Dickson mostly just lists results by various a
Minimal polynomial of 2cos(2pi/n)
For an integer , the minimal polynomial of is the non-zero monic polynomial of degree for and degree for with integer coefficients, such that . Here denotes the Euler's totient function. In particular
Non-integer base of numeration
A non-integer representation uses non-integer numbers as the radix, or base, of a positional numeral system. For a non-integer radix β > 1, the value of is The numbers di are non-negative integers les
Cobham's theorem
Cobham's theorem is a theorem in combinatorics on words that has important connections with number theory, notably transcendental numbers, and automata theory. Informally, the theorem gives the condit
Covering set
In mathematics, a covering set for a sequence of integers refers to a set of prime numbers such that every term in the sequence is divisible by at least one member of the set. The term "covering set"
Computational number theory
In mathematics and computer science, computational number theory, also known as algorithmic number theory, is the study of computational methods for investigating and solving problems in number theory
P-adic gamma function
In mathematics, the p-adic gamma function Γp is a function of a p-adic variable analogous to the gamma function. It was first explicitly defined by , though pointed out that implicitly used the same f
Contou-Carrère symbol
In mathematics, the Contou-Carrère symbol 〈a,b〉 is a Steinberg symbol defined on pairs of invertible elements of the ring of Laurent power series over an Artinian ring k, taking values in the group of
Elkies trinomial curves
In number theory, the Elkies trinomial curves are certain hyperelliptic curves constructed by Noam Elkies which have the property that rational points on them correspond to trinomial polynomials givin
Sidon sequence
In number theory, a Sidon sequence is a sequence of natural numbers in which all pairwise sums (for ) are different. Sidon sequences are also called Sidon sets; they are named after the Hungarian math
Knödel number
In number theory, an n-Knödel number for a given positive integer n is a composite number m with the property that each i < m coprime to m satisfies . The concept is named after Walter Knödel. The set
Hodge–Tate module
In mathematics, a Hodge–Tate module is an analogue of a Hodge structure over p-adic fields. Serre introduced and named Hodge–Tate structures using the results of Tate on p-divisible groups.
Dubner's conjecture
Dubner's conjecture is an as yet (2018) unsolved conjecture by American mathematician Harvey Dubner. It states that every even number greater than 4208 is the sum of two t-primes, where a t-prime is a
Fourth power
In arithmetic and algebra, the fourth power of a number n is the result of multiplying four instances of n together. So: n4 = n × n × n × n Fourth powers are also formed by multiplying a number by its
Sylvester's sequence
In number theory, Sylvester's sequence is an integer sequence in which each term of the sequence is the product of the previous terms, plus one. The first few terms of the sequence are 2, 3, 7, 43, 18
Tate's algorithm
In the theory of elliptic curves, Tate's algorithm takes as input an of an elliptic curve E over , or more generally an algebraic number field, and a prime or prime ideal p. It returns the exponent fp
Fueter–Pólya theorem
The Fueter–Pólya theorem, first proved by Rudolf Fueter and George Pólya, states that the only quadratic polynomial pairing functions are the Cantor polynomials.
P-adic analysis
In mathematics, p-adic analysis is a branch of number theory that deals with the mathematical analysis of functions of p-adic numbers. The theory of complex-valued numerical functions on the p-adic nu
Coprime integers
In mathematics, two integers a and b are coprime, relatively prime or mutually prime if the only positive integer that is a divisor of both of them is 1. Consequently, any prime number that divides a
Prime omega function
In number theory, the prime omega functions and count the number of prime factors of a natural number Thereby (little omega) counts each distinct prime factor, whereas the related function (big omega)
Primefree sequence
In mathematics, a primefree sequence is a sequence of integers that does not contain any prime numbers. More specifically, it usually means a sequence defined by the same recurrence relation as the Fi
Random Fibonacci sequence
In mathematics, the random Fibonacci sequence is a stochastic analogue of the Fibonacci sequence defined by the recurrence relation , where the signs + or − are chosen at random with equal probability
Weil–Châtelet group
In arithmetic geometry, the Weil–Châtelet group or WC-group of an algebraic group such as an abelian variety A defined over a field K is the abelian group of principal homogeneous spaces for A, define
List of recreational number theory topics
This is a list of recreational number theory topics (see number theory, recreational mathematics). Listing here is not pejorative: many famous topics in number theory have origins in challenging probl
Noncommutative unique factorization domain
In mathematics, a noncommutative unique factorization domain is a noncommutative ring with the unique factorization property.
Lone divider
The lone divider procedure is a procedure for proportional cake-cutting. It involves a heterogenous and divisible resource, such as a birthday cake, and n partners with different preferences over diff
Skewes's number
In number theory, Skewes's number is any of several large numbers used by the South African mathematician Stanley Skewes as upper bounds for the smallest natural number for which where π is the prime-
Hecke algebra
In mathematics, the Hecke algebra is the algebra generated by Hecke operators.
Durfee square
In number theory, a Durfee square is an attribute of an integer partition. A partition of n has a Durfee square of size s if s is the largest number such that the partition contains at least s parts w
Arithmetic group
In mathematics, an arithmetic group is a group obtained as the integer points of an algebraic group, for example They arise naturally in the study of arithmetic properties of quadratic forms and other
Harmonic divisor number
In mathematics, a harmonic divisor number, or Ore number (named after Øystein Ore who defined it in 1948), is a positive integer whose divisors have a harmonic mean that is an integer. The first few h
Fifth power (algebra)
In arithmetic and algebra, the fifth power or sursolid of a number n is the result of multiplying five instances of n together: n5 = n × n × n × n × n. Fifth powers are also formed by multiplying a nu
Abel's summation formula
In mathematics, Abel's summation formula, introduced by Niels Henrik Abel, is intensively used in analytic number theory and the study of special functions to compute series.
In mathematics, the remainder is the amount "left over" after performing some computation. In arithmetic, the remainder is the integer "left over" after dividing one integer by another to produce an i
Champernowne constant
In mathematics, the Champernowne constant C10 is a transcendental real constant whose decimal expansion has important properties. It is named after economist and mathematician D. G. Champernowne, who
Hodge–Arakelov theory
In mathematics, Hodge–Arakelov theory of elliptic curves is an analogue of classical and p-adic Hodge theory for elliptic curves carried out in the framework of Arakelov theory. It was introduced by M
Gregory coefficients
Gregory coefficients Gn, also known as reciprocal logarithmic numbers, Bernoulli numbers of the second kind, or Cauchy numbers of the first kind, are the rational numbers that occur in the Maclaurin s
Character group
In mathematics, a character group is the group of representations of a group by complex-valued functions. These functions can be thought of as one-dimensional matrix representations and so are special
Haran's diamond theorem
In mathematics, the Haran diamond theorem gives a general sufficient condition for a separable extension of a Hilbertian field to be Hilbertian.
P-adic Teichmüller theory
In mathematics, p-adic Teichmüller theory describes the "uniformization" of p-adic curves and their moduli, generalizing the usual Teichmüller theory that describes the uniformization of Riemann surfa
In number theory and combinatorics, a multipartition of a positive integer n is a way of writing n as a sum, each element of which is in turn a partition. The concept is also found in the theory of Li
Paley graph
In mathematics, Paley graphs are dense undirected graphs constructed from the members of a suitable finite field by connecting pairs of elements that differ by a quadratic residue. The Paley graphs fo
Artin–Hasse exponential
In mathematics, the Artin–Hasse exponential, introduced by Artin and Hasse, is the power series given by
Brjuno number
In mathematics, a Brjuno number is a special type of irrational number.
New York Number Theory Seminar
The New York Number Theory Seminar is a research seminar devoted to the theory of numbers and related parts of mathematics and physics. The seminar began in 1982 under the founding organizers Harvey C
In arithmetic geometry, a Frobenioid is a category with some extra structure that generalizes the theory of line bundles on models of finite extensions of global fields. Frobenioids were introduced by
Harmonic number
In mathematics, the n-th harmonic number is the sum of the reciprocals of the first n natural numbers: Starting from n = 1, the sequence of harmonic numbers begins: Harmonic numbers are related to the
Moessner's theorem
In number theory, Moessner's theorem or Moessner's magicis related to an arithmetical algorithm to produce an infinite sequence of the exponents of positive integers with by recursively manipulating t
Cyclic number
A cyclic number is an integer for which cyclic permutations of the digits are successive integer multiples of the number. The most widely known is the six-digit number 142857, whose first six integer
Probabilistic number theory
In mathematics, Probabilistic number theory is a subfield of number theory, which explicitly uses probability to answer questions about the integers and integer-valued functions. One basic idea underl
Bernoulli polynomials
In mathematics, the Bernoulli polynomials, named after Jacob Bernoulli, combine the Bernoulli numbers and binomial coefficients. They are used for series expansion of functions, and with the Euler–Mac
Arithmetic topology
Arithmetic topology is an area of mathematics that is a combination of algebraic number theory and topology. It establishes an analogy between number fields and closed, orientable 3-manifolds.
Ring of modular forms
In mathematics, the ring of modular forms associated to a subgroup Γ of the special linear group SL(2, Z) is the graded ring generated by the modular forms of Γ. The study of rings of modular forms de
Arithmetic hyperbolic 3-manifold
In mathematics, more precisely in group theory and hyperbolic geometry, Arithmetic Kleinian groups are a special class of Kleinian groups constructed using orders in quaternion algebras. They are part
Bertrand's postulate
In number theory, Bertrand's postulate is a theorem stating that for any integer , there always exists at least one prime number with A less restrictive formulation is: for every , there is always at
Arithmetic zeta function
In mathematics, the arithmetic zeta function is a zeta function associated with a scheme of finite type over integers. The arithmetic zeta function generalizes the Riemann zeta function and Dedekind z
Basel problem
The Basel problem is a problem in mathematical analysis with relevance to number theory, concerning an infinite sum of inverse squares. It was first posed by Pietro Mengoli in 1650 and solved by Leonh
Stoneham number
In mathematics, the Stoneham numbers are a certain class of real numbers, named after mathematician (1920–1996). For coprime numbers b, c > 1, the Stoneham number αb,c is defined as It was shown by St
Brandt matrix
In mathematics, Brandt matrices are matrices, introduced by Brandt, that are related to the number of ideals of given norm in an ideal class of a definite quaternion algebra over the rationals, and th
Dirichlet hyperbola method
In number theory, the Dirichlet hyperbola method is a technique to evaluate the sum where are multiplicative functions with , where is the Dirichlet convolution. It uses the fact that
Jacobsthal sum
In mathematics, Jacobsthal sums are finite sums of Legendre symbols related to Gauss sums. They were introduced by Jacobsthal.
Number theory
Number theory (or arithmetic or higher arithmetic in older usage) is a branch of pure mathematics devoted primarily to the study of the integers and integer-valued functions. German mathematician Carl
Bost–Connes system
In mathematics, a Bost–Connes system is a quantum statistical dynamical system related to an algebraic number field, whose partition function is related to the Dedekind zeta function of the number fie
Solenoid (mathematics)
In mathematics, a solenoid is a compact connected topological space (i.e. a continuum) that may be obtained as the inverse limit of an inverse system of topological groups and continuous homomorphisms
Residue-class-wise affine group
In mathematics, specifically in group theory, residue-class-wise affinegroups are certain permutation groups acting on (the integers), whose elements are bijectiveresidue-class-wise affine mappings. A
Persistence of a number
In mathematics, the persistence of a number is the number of times one must apply a given operation to an integer before reaching a fixed point at which the operation no longer alters the number. Usua
Ruler function
In number theory, the ruler function of an integer can be either of two closely-related functions. One of these functions counts the number of times can be evenly divided by two, which for the numbers
Glaisher–Kinkelin constant
In mathematics, the Glaisher–Kinkelin constant or Glaisher's constant, typically denoted A, is a mathematical constant, related to the K-function and the Barnes G-function. The constant appears in a n
Cole Prize
The Frank Nelson Cole Prize, or Cole Prize for short, is one of twenty-two prizes awarded to mathematicians by the American Mathematical Society, one for an outstanding contribution to algebra, and th
Square number
In mathematics, a square number or perfect square is an integer that is the square of an integer; in other words, it is the product of some integer with itself. For example, 9 is a square number, sinc
Change-making problem
The change-making problem addresses the question of finding the minimum number of coins (of certain denominations) that add up to a given amount of money. It is a special case of the integer knapsack
Farey sequence
In mathematics, the Farey sequence of order n is the sequence of completely reduced fractions, either between 0 and 1, or without this restriction, which when in lowest terms have denominators less th
Kaprekar number
In mathematics, a natural number in a given number base is a -Kaprekar number if the representation of its square in that base can be split into two parts, where the second part has digits, that add u
Selmer group
In arithmetic geometry, the Selmer group, named in honor of the work of Ernst Sejersted Selmer by John William Scott Cassels, is a group constructed from an isogeny of abelian varieties.
Iwasawa algebra
In mathematics, the Iwasawa algebra Λ(G) of a profinite group G is a variation of the group ring of G with p-adic coefficients that take the topology of G into account. More precisely, Λ(G) is the inv
Cabtaxi number
In mathematics, the n-th cabtaxi number, typically denoted Cabtaxi(n), is defined as the smallest positive integer that can be written as the sum of two positive or negative or 0 cubes in n ways. Such