• List of recreational number theory topics
• Topics in cryptography

Factors

• Composite number
  Composite number
  A composite number is a positive integer which has a positive divisor other than one or itself. In other words a composite number is any positive integer greater than one that is not a prime number....
  
  
  • Highly composite number
    Highly composite number
    A highly composite number is a positive integer with more divisors than any positive integer smaller than itself.The initial or smallest twenty-one highly composite numbers are listed in the table at right....
    
    
• Even and odd numbers
  Even and odd numbers
  In mathematics, the parity of an object states whether it is even or odd.This concept begins with integers. An even number is an integer that is "evenly divisible" by 2, i.e., divisible by 2 without remainder; an odd number is an integer that is not evenly divisible by 2...
  
  
  • Parity
• Divisor
  Divisor
  In mathematics, a divisor of an integer n, also called a factor of n, is an integer which divides n without leaving a remainder.-Explanation:...
  
  , aliquot part
  • Greatest common divisor
    Greatest common divisor
    In mathematics, the greatest common divisor , also known as the greatest common factor , or highest common factor , of two or more non-zero integers, is the largest positive integer that divides the numbers without a remainder.For example, the GCD of 8 and 12 is 4.This notion can be extended to...
    
    
  • Least common multiple
    Least common multiple
    In arithmetic and number theory, the least common multiple of two integers a and b, usually denoted by LCM, is the smallest positive integer that is a multiple of both a and b...
    
    
  • Euclidean algorithm
    Euclidean algorithm
    In mathematics, the Euclidean algorithm is an efficient method for computing the greatest common divisor of two integers, also known as the greatest common factor or highest common factor...
    
    
  • Coprime
    Coprime
    In number theory, a branch of mathematics, two integers a and b are said to be coprime or relatively prime if the only positive integer that evenly divides both of them is 1. This is the same thing as their greatest common divisor being 1...
    
    
  • Euclid's lemma
    Euclid's lemma
    In mathematics, Euclid's lemma is an important lemma regarding divisibility and prime numbers. In its simplest form, the lemma states that a prime number that divides a product of two integers must divide one of the two integers...
    
    
  • Bézout's identity
    Bézout's identity
    In number theory, Bézout's identity for two integers a, b is an expressionwhere x and y are integers , such that d is a common divisor of a and b. Bézout's lemma states that such coefficients exist for every pair of nonzero integers...
    
    , Bézout's lemma
  • Extended Euclidean algorithm
    Extended Euclidean algorithm
    The extended Euclidean algorithm is an extension to the Euclidean algorithm. Besides finding the greatest common divisor of integers a and b, as the Euclidean algorithm does, it also finds integers x and y that satisfy Bézout's identityThe extended Euclidean algorithm is particularly useful when a...
    
    
  • Table of divisors
    Table of divisors
    The tables below list all of the divisors of the numbers 1 to 1000.A divisor of an integer n is an integer m, say, for which n/m is again an integer . For example, 3 is a divisor of 21, since 21/3 = 7 .If m is a divisor of n then so is −m...
    
    
• Prime number
  Prime number
  A prime number is a natural number greater than 1 that has no positive divisors other than 1 and itself. A natural number greater than 1 that is not a prime number is called a composite number. For example 5 is prime, as only 1 and 5 divide it, whereas 6 is composite, since it has the divisors 2...
  
  , prime power
  Prime power
  In mathematics, a prime power is a positive integer power of a prime number.For example: 5=51, 9=32 and 16=24 are prime powers, while6=2×3, 15=3×5 and 36=62=22×32 are not...
  
  
  • Bonse's inequality
    Bonse's inequality
    In number theory, Bonse's inequality, named after H. Bonse, states that if p1, ..., pn, pn+1 are the smallest n + 1 prime numbers and n ≥ 4, then...
    
    
• Prime factor
  Prime factor
  In number theory, the prime factors of a positive integer are the prime numbers that divide that integer exactly, without leaving a remainder. The process of finding these numbers is called integer factorization, or prime factorization. A prime factor can be visualized by understanding Euclid's...
  
  
  • Table of prime factors
    Table of prime factors
    The tables contain the prime factorization of the natural numbers from 1 to 1000.When n is a prime number, the prime factorization is just n itself, written in bold below.The number 1 is called a unit...
    
    
• Formula for primes
  Formula for primes
  In number theory, a formula for primes is a formula generating the prime numbers, exactly and without exception. No such formula which is easily computable is presently known...
  
  
• Factorization
  Factorization
  In mathematics, factorization or factoring is the decomposition of an object into a product of other objects, or factors, which when multiplied together give the original...
  
  
  • RSA number
• Fundamental theorem of arithmetic
  Fundamental theorem of arithmetic
  In number theory, the fundamental theorem of arithmetic states that any integer greater than 1 can be written as a unique product of prime numbers...
  
  
• Square-free
  Square-free
  In mathematics, an element r of a unique factorization domain R is called square-free if it is not divisible by a non-trivial square. That is, every s such that s^2\mid r is a unit of R....
  
  
  • Square-free integer
    Square-free integer
    In mathematics, a square-free, or quadratfrei, integer is one divisible by no perfect square, except 1. For example, 10 is square-free but 18 is not, as it is divisible by 9 = 32...
    
    
  • Square-free polynomial
    Square-free polynomial
    In mathematics, a square-free polynomial is a polynomial with no square factors, i.e, f \in F[x] is square-free if and only if b^2 \nmid f for every b \in F[x] with non-zero degree. This definition implies that no factors of higher order can exist, either, for if b3 divided the polynomial, then b2...
    
    
• Square number
  Square number
  In mathematics, a square number, sometimes also called a perfect square, is an integer that is the square of an integer; in other words, it is the product of some integer with itself...
  
  
• Power of two
  Power of two
  In mathematics, a power of two means a number of the form 2n where n is an integer, i.e. the result of exponentiation with as base the number two and as exponent the integer n....
  
  
• Integer-valued polynomial
  Integer-valued polynomial
  In mathematics, an integer-valued polynomial P is a polynomial taking an integer value P for every integer n. Certainly every polynomial with integer coefficients is integer-valued. There are simple examples to show that the converse is not true: for example the polynomialgiving the triangle...
  
  

Fraction

Fraction (mathematics)

A fraction represents a part of a whole or, more generally, any number of equal parts. When spoken in everyday English, we specify how many parts of a certain size there are, for example, one-half, five-eighths and three-quarters.A common or "vulgar" fraction, such as 1/2, 5/8, 3/4, etc., consists...

s

• Rational number
  Rational number
  In mathematics, a rational number is any number that can be expressed as the quotient or fraction a/b of two integers, with the denominator b not equal to zero. Since b may be equal to 1, every integer is a rational number...
  
  
• Unit fraction
  Unit fraction
  A unit fraction is a rational number written as a fraction where the numerator is one and the denominator is a positive integer. A unit fraction is therefore the reciprocal of a positive integer, 1/n...
  
  
• Irreducible fraction
  Irreducible fraction
  An irreducible fraction is a vulgar fraction in which the numerator and denominator are smaller than those in any other equivalent vulgar fraction...
  
   = in lowest terms
• Dyadic fraction
• Recurring decimal
• Cyclic number
  Cyclic number
  A cyclic number is an integer in which cyclic permutations of the digits are successive multiples of the number. The most widely known is 142857:For example:Multiples of these fractions exhibit cyclic permutation:...
  
  
• Farey sequence
  Farey sequence
  In mathematics, the Farey sequence of order n is the sequence of completely reduced fractions between 0 and 1 which, when in lowest terms, have denominators less than or equal to n, arranged in order of increasing size....
  
  
  • Ford circle
    Ford circle
    In mathematics, a Ford circle is a circle with centre at and radius 1/, where p/q is an irreducible fraction, i.e. p and q are coprime integers...
    
    
  • Stern–Brocot tree
• Dedekind sum
  Dedekind sum
  In mathematics, Dedekind sums, named after Richard Dedekind, 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 of the Dedekind eta function. They have subsequently been much...
  
  
• Egyptian fraction

Modular arithmetic

Modular arithmetic

In mathematics, modular arithmetic is a system of arithmetic for integers, where numbers "wrap around" after they reach a certain value—the modulus....

• Montgomery reduction
  Montgomery reduction
  In arithmetic computation, Montgomery reduction is an algorithm introduced in 1985 by Peter Montgomery that allows modular arithmetic to be performed efficiently when the modulus is large ....
  
  
• Modular exponentiation
  Modular exponentiation
  Modular exponentiation is a type of exponentiation performed over a modulus. It is particularly useful in computer science, especially in the field of cryptography....
  
  
• Linear congruence theorem
• Method of successive substitution
  Method of successive substitution
  In modular arithmetic, the method of successive substitution is a method of solving problems of simultaneous congruences by using the definition of the congruence equation...
  
  
• Chinese remainder theorem
  Chinese remainder theorem
  The Chinese remainder theorem is a result about congruences in number theory and its generalizations in abstract algebra.In its most basic form it concerned with determining n, given the remainders generated by division of n by several numbers...
  
  
• Fermat's little theorem
  Fermat's little theorem
  Fermat's little theorem states that if p is a prime number, then for any integer a, a p − a will be evenly divisible by p...
  
  
  • Proofs of Fermat's little theorem
    Proofs of Fermat's little theorem
    This article collects together a variety of proofs of Fermat's little theorem, which states thata^p \equiv a \pmod p \,\!for every prime number p and every integer a .-Simplifications:...
    
    
• Fermat quotient
• Euler's totient function
  Euler's totient function
  In number theory, the totient \varphi of a positive integer n is defined to be the number of positive integers less than or equal to n that are coprime to n In number theory, the totient \varphi(n) of a positive integer n is defined to be the number of positive integers less than or equal to n that...
  
  
  • Noncototient
    Noncototient
    In mathematics, a noncototient is a positive integer n that cannot be expressed as the difference between a positive integer m and the number of coprime integers below it. That is, m − φ = n, where φ stands for Euler's totient function, has no solution for m...
    
    
  • Nontotient
    Nontotient
    In number theory, a nontotient is a positive integer n which is not in the range of Euler's totient function φ, that is, for which φ = n has no solution. In other words, n is a nontotient if there is no integer x that has exactly n coprimes below it. All odd numbers are nontotients, except 1,...
    
    
• Euler's theorem
• Wilson's theorem
• Primitive root modulo n
  Primitive root modulo n
  In modular arithmetic, a branch of number theory, a primitive root modulo n is any number g with the property that any number coprime to n is congruent to a power of g modulo n. In other words, g is a generator of the multiplicative group of integers modulo n...
  
  
  • Multiplicative order
    Multiplicative order
    In number theory, given an integer a and a positive integer n with gcd = 1, the multiplicative order of a modulo n is the smallest positive integer k withThe order of a modulo n is usually written ordn, or On.- Example :To determine the multiplicative order of 4 modulo 7, we compute 42 = 16 ≡ 2 ...
    
    
  • Discrete logarithm
    Discrete logarithm
    In mathematics, specifically in abstract algebra and its applications, discrete logarithms are group-theoretic analogues of ordinary logarithms. In particular, an ordinary logarithm loga is a solution of the equation ax = b over the real or complex numbers...
    
    
• Quadratic residue
  • Euler's criterion
    Euler's criterion
    In mathematics, Euler's criterion is used in determining in number theory whether a given integer is a quadratic residue modulo a prime.-Definition:Euler's criterion states:Let p be an odd prime and a an integer coprime to p. Then...
    
    
  • Legendre symbol
    Legendre symbol
    In number theory, the Legendre symbol is a multiplicative function with values 1, −1, 0 that is a quadratic character modulo a prime number p: its value on a quadratic residue mod p is 1 and on a quadratic non-residue is −1....
    
    
  • Gauss's lemma (number theory)
    Gauss's lemma (number theory)
    Gauss's lemma in number theory gives a condition for an integer to be a quadratic residue. Although it is not useful computationally, it has theoretical significance, being involved in some proofs of quadratic reciprocity....
    
    
• Congruence of squares
  Congruence of squares
  In number theory, a congruence of squares is a congruence commonly used in integer factorization algorithms.-Derivation:Given a positive integer n, Fermat's factorization method relies on finding numbers x, y satisfying the equality...
  
  
• Luhn formula
• Mod n cryptanalysis
  Mod n cryptanalysis
  In cryptography, mod n cryptanalysis is an attack applicable to block and stream ciphers. It is a form of partitioning cryptanalysis that exploits unevenness in how the cipher operates over equivalence classes modulo n...
  
  

Arithmetic function

Arithmetic function

In number theory, an arithmetic function is a real or complex valued function ƒ defined on the set of natural numbers In number theory, an arithmetic (or arithmetical) function is a real or complex valued function ƒ(n) defined on the set of natural numbers In number theory, an arithmetic (or...

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• Multiplicative function
  Multiplicative function
  In number theory, a multiplicative function is an arithmetic function f of the positive integer n with the property that f = 1 and whenevera and b are coprime, then...
  
  
• Additive function
  Additive function
  In mathematics the term additive function has two different definitions, depending on the specific field of application.In algebra an additive function is a function that preserves the addition operation:for any two elements x and y in the domain. For example, any linear map is additive...
  
  
• Dirichlet convolution
  Dirichlet convolution
  In mathematics, the Dirichlet convolution is a binary operation defined for arithmetic functions; it is important in number theory. It was developed by Johann Peter Gustav Lejeune Dirichlet, a German mathematician.-Definition:...
  
  
• Erdős–Kac theorem
  Erdos–Kac theorem
  In number theory, the Erdős–Kac theorem, named after Paul Erdős and Mark Kac, and also known as the fundamental theorem of probabilistic number theory, states that if ω is the number of distinct prime factors of n, then, loosely speaking, the probability distribution ofis the standard normal...
  
  
• Möbius function
  Möbius function
  The classical Möbius function μ is an important multiplicative function in number theory and combinatorics. The German mathematician August Ferdinand Möbius introduced it in 1832...
  
  
  • Möbius inversion formula
    Möbius inversion formula
    In mathematics, the classic Möbius inversion formula was introduced into number theory during the 19th century by August Ferdinand Möbius. Other Möbius inversion formulas are obtained when different local finite partially ordered sets replace the classic case of the natural numbers ordered by...
    
    
• Divisor function
  Divisor function
  In mathematics, and specifically in number theory, a divisor function is an arithmetical function related to the divisors of an integer. When referred to as the divisor function, it counts the number of divisors of an integer. It appears in a number of remarkable identities, including relationships...
  
  
• Liouville function
• Partition function (number theory)
  • Integer partition
  • Bell numbers
  • Landau's function
    Landau's function
    In mathematics, Landau's function g, named after Edmund Landau, is defined for every natural number n to be the largest order of an element of the symmetric group Sn...
    
    
  • Pentagonal number theorem
• Bell series
  Bell series
  In mathematics, the Bell series is a formal power series used to study properties of arithmetical functions. Bell series were introduced and developed by Eric Temple Bell....
  
  
• Lambert series

Analytic number theory

Analytic number theory

In mathematics, analytic number theory is a branch of number theory that uses methods from mathematical analysis to solve problems about the integers. It is often said to have begun with Dirichlet's introduction of Dirichlet L-functions to give the first proof of Dirichlet's theorem on arithmetic...

: additive problems

• Twin prime
  Twin prime
  A twin prime is a prime number that differs from another prime number by two. Except for the pair , this is the smallest possible difference between two primes. Some examples of twin prime pairs are , , , , and...
  
  
  • Brun's constant
• Cousin prime
  Cousin prime
  In mathematics, cousin primes are prime numbers that differ by four; compare this with twin primes, pairs of prime numbers that differ by two, and sexy primes, pairs of prime numbers that differ by six....
  
  
• Prime quadruplet
• Sexy prime
  Sexy prime
  In mathematics, a sexy prime is a prime number that differs from another prime number by six. For example, the numbers 5 and 11 are both sexy primes, because they differ by 6...
  
  
• Sophie Germain prime
  Sophie Germain prime
  In number theory, a prime number p is a Sophie Germain prime if 2p + 1 is also prime. For example, 23 is a Sophie Germain prime because it is a prime and 2 × 23 + 1 = 47, and 47 is also a prime number...
  
  
• Cunningham chain
  Cunningham chain
  In mathematics, a Cunningham chain is a certain sequence of prime numbers. Cunningham chains are named after mathematician A. J. C. Cunningham. They are also called chains of nearly doubled primes....
  
  
• Goldbach's conjecture
  Goldbach's conjecture
  Goldbach's conjecture is one of the oldest unsolved problems in number theory and in all of mathematics. It states:A Goldbach number is a number that can be expressed as the sum of two odd primes...
  
  
  • Goldbach's weak conjecture
    Goldbach's weak conjecture
    In number theory, Goldbach's weak conjecture, also known as the odd Goldbach conjecture, the ternary Goldbach problem, or the 3-primes problem, states that:...
    
    
• Second Hardy–Littlewood conjecture
• Hardy–Littlewood circle method
• Schinzel's hypothesis H
  Schinzel's hypothesis H
  In mathematics, Schinzel's hypothesis H is a very broad generalisation of conjectures such as the twin prime conjecture. It aims to define the possible scope of a conjecture of the nature that several sequences of the type...
  
  
• Bateman–Horn conjecture
• Waring's problem
  Waring's problem
  In number theory, Waring's problem, proposed in 1770 by Edward Waring, asks whether for every natural number k there exists an associated positive integer s such that every natural number is the sum of at most s kth powers of natural numbers...
  
  
  • Brahmagupta–Fibonacci identity
  • Euler's four-square identity
    Euler's four-square identity
    In mathematics, Euler's four-square identity says that the product of two numbers, each of which being a sum of four squares, is itself a sum of four squares. Specifically:=\,...
    
    
  • Lagrange's four-square theorem
    Lagrange's four-square theorem
    Lagrange's four-square theorem, also known as Bachet's conjecture, states that any natural number can be represented as the sum of four integer squaresp = a_0^2 + a_1^2 + a_2^2 + a_3^2\ where the four numbers are integers...
    
    
  • Taxicab number
    Taxicab number
    In mathematics, the nth taxicab number, typically denoted Ta or Taxicab, is defined as the smallest number that can be expressed as a sum of two positive algebraic cubes in n distinct ways. The concept was first mentioned in 1657 by Bernard Frénicle de Bessy, and was made famous in the early 20th...
    
    
  • Generalized taxicab number
    Generalized taxicab number
    In mathematics, the generalized taxicab number Taxicab is the smallest number which can be expressed as the sum of j kth positive powers in n different ways...
    
    
• Cabtaxi number
  Cabtaxi number
  In mathematics, the n-th cabtaxi number, typically denoted Cabtaxi, 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...
  
  
• Schnirelmann density
  Schnirelmann density
  In additive number theory, the Schnirelmann density of a sequence of numbers is a way to measure how "dense" the sequence is. It is named after Russian mathematician L.G...
  
  
• Sumset
  Sumset
  In additive combinatorics, the sumset of two subsets A and B of an abelian group G is defined to be the set of all sums of an element from A with an element from B...
  
  
• Landau–Ramanujan constant
• Sierpinski number
  • Seventeen or Bust
    Seventeen or Bust
    Seventeen or Bust is a distributed computing project started in March 2002 to solve the last seventeen cases in the Sierpinski problem.-Goals:...
    
    
• Niven's constant
  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"...
  
  

Quadratic form

Quadratic form

In mathematics, a quadratic form is a homogeneous polynomial of degree two in a number of variables. For example,4x^2 + 2xy - 3y^2\,\!is a quadratic form in the variables x and y....

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• Unimodular lattice
  Unimodular lattice
  In mathematics, a unimodular lattice is a lattice of determinant 1 or −1.The E8 lattice and the Leech lattice are two famous examples.- Definitions :...
  
  
• Fermat's theorem on sums of two squares
  • Proofs of Fermat's theorem on sums of two squares
    Proofs of Fermat's theorem on sums of two squares
    Fermat's theorem on sums of two squares asserts that an odd prime number p can be expressed aswith integer x and y if and only if p is congruent to 1 . The statement was announced by Fermat in 1640, but he supplied no proof....
    
    

L-function

L-function

The theory of L-functions has become a very substantial, and still largely conjectural, part of contemporary analytic number theory. In it, broad generalisations of the Riemann zeta function and the L-series for a Dirichlet character are constructed, and their general properties, in most cases...

s

• Riemann zeta function
  • Basel problem
    Basel problem
    The Basel problem is a famous problem in mathematical analysis with relevance to number theory, first posed by Pietro Mengoli in 1644 and solved by Leonhard Euler in 1735. Since the problem had withstood the attacks of the leading mathematicians of the day, Euler's solution brought him immediate...
    
     on ζ(2)
  • Hurwitz zeta function
  • Bernoulli number
    Bernoulli number
    In mathematics, the Bernoulli numbers Bn are a sequence of rational numbers with deep connections to number theory. They are closely related to the values of the Riemann zeta function at negative integers....
    
    
    • Agoh–Giuga conjecture
    • Von Staudt–Clausen theorem
      Von Staudt–Clausen theorem
      In number theory, the von Staudt–Clausen theorem is a result determining the fractional part of Bernoulli numbers, found independently by and ....
      
      
• Dirichlet series
• Euler product
  Euler product
  In number theory, an Euler product is an expansion of a Dirichlet series into an infinite product indexed by prime numbers. The name arose from the case of the Riemann zeta-function, where such a product representation was proved by Leonhard Euler.-Definition:...
  
  
• Prime number theorem
  Prime number theorem
  In number theory, the prime number theorem describes the asymptotic distribution of the prime numbers. The prime number theorem gives a general description of how the primes are distributed amongst the positive integers....
  
  
  • Prime-counting function
  • Offset logarithmic integral
  • Legendre's constant
    Legendre's constant
    Legendre's constant is a mathematical constant occurring in a formula conjectured by Adrien-Marie Legendre to capture the asymptotic behavior of the prime-counting function \scriptstyle\pi...
    
    
  • Skewes' number
  • Bertrand's postulate
    • Proof of Bertrand's postulate
    • Proof that the sum of the reciprocals of the primes diverges
  • Cramér's conjecture
• Riemann hypothesis
  Riemann hypothesis
  In mathematics, the Riemann hypothesis, proposed by , is a conjecture about the location of the zeros of the Riemann zeta function which states that all non-trivial zeros have real part 1/2...
  
  
  • Critical line theorem
  • Hilbert–Pólya conjecture
  • Generalized Riemann hypothesis
    Generalized Riemann hypothesis
    The Riemann hypothesis is one of the most important conjectures in mathematics. It is a statement about the zeros of the Riemann zeta function. Various geometrical and arithmetical objects can be described by so-called global L-functions, which are formally similar to the Riemann zeta-function...
    
    
  • Mertens function, Mertens conjecture
    Mertens conjecture
    In mathematics, the Mertens conjecture is the incorrect statement that the Mertens function M is bounded by √n, which implies the Riemann hypothesis...
    
    , Meissel–Mertens constant
  • De Bruijn–Newman constant
• Dirichlet character
  Dirichlet character
  In number theory, Dirichlet characters are certain arithmetic functions which arise from completely multiplicative characters on the units of \mathbb Z / k \mathbb Z...
  
  
• Dirichlet L-series
  • Siegel zero
    Siegel zero
    In mathematics, more specifically in the field of analytic number theory, a Siegel zero, named after Carl Ludwig Siegel, is a type of potential counterexample to the generalized Riemann hypothesis, on the zeroes of Dirichlet L-function....
    
    
• Dirichlet's theorem on arithmetic progressions
  Dirichlet's theorem on arithmetic progressions
  In number theory, Dirichlet's theorem, also called the Dirichlet prime number theorem, states that for any two positive coprime integers a and d, there are infinitely many primes of the form a + nd, where n ≥ 0. In other words, there are infinitely many primes which are...
  
  
  • Linnik's theorem
    Linnik's theorem
    Linnik's theorem in analytic number theory answers a natural question after Dirichlet's theorem on arithmetic progressions. It asserts that, if we denote p the least prime in the arithmetic progressiona + nd,\...
    
    
  • Elliott–Halberstam conjecture
• Functional equation (L-function)
  Functional equation (L-function)
  In mathematics, the L-functions of number theory are expected to have several characteristic properties, one of which is that they satisfy certain functional equations. There is an elaborate theory of what these equations should be, much of which is still conjectural...
  
  
• Chebotarev's density theorem
  Chebotarev's density theorem
  Chebotarev's density theorem in algebraic number theory describes statistically the splitting of primes in a given Galois extension K of the field Q of rational numbers. Generally speaking, a prime integer will factor into several ideal primes in the ring of algebraic integers of K. There are only...
  
  
• Local zeta function
  • Weil conjectures
    Weil conjectures
    In mathematics, the Weil conjectures were some highly-influential proposals by on the generating functions derived from counting the number of points on algebraic varieties over finite fields....
    
    
• Modular form
  Modular form
  In mathematics, a modular form is a analytic function on the upper half-plane satisfying a certain kind of functional equation and growth condition. The theory of modular forms therefore belongs to complex analysis but the main importance of the theory has traditionally been in its connections...
  
  
  • modular group
  • Congruence subgroup
    Congruence subgroup
    In mathematics, a congruence subgroup of a matrix group with integer entries is a subgroup defined by congruence conditions on the entries. A very simple example would be invertible 2x2 integer matrices of determinant 1, such that the off-diagonal entries are even.An importance class of congruence...
    
    
  • Hecke operator
    Hecke operator
    In mathematics, in particular in the theory of modular forms, a Hecke operator, studied by , is a certain kind of "averaging" operator that plays a significant role in the structure of vector spaces of modular forms and more general automorphic representations....
    
    
  • Cusp form
    Cusp form
    In number theory, a branch of mathematics, a cusp form is a particular kind of modular form, distinguished in the case of modular forms for the modular group by the vanishing in the Fourier series expansion \Sigma a_n q^n...
    
    
  • Eisenstein series
    Eisenstein series
    Eisenstein series, named after German mathematician Gotthold Eisenstein, are particular modular forms with infinite series expansions that may be written down directly...
    
    
  • Modular curve
    Modular curve
    In number theory and algebraic geometry, a modular curve Y is a Riemann surface, or the corresponding algebraic curve, constructed as a quotient of the complex upper half-plane H by the action of a congruence subgroup Γ of the modular group of integral 2×2 matrices SL...
    
    
  • Ramanujan–Petersson conjecture
• Birch and Swinnerton-Dyer conjecture
  Birch and Swinnerton-Dyer conjecture
  In mathematics, the Birch and Swinnerton-Dyer conjecture is an open problem in the field of number theory. Its status as one of the most challenging mathematical questions has become widely recognized; the conjecture was chosen as one of the seven Millennium Prize Problems listed by the Clay...
  
  
• Automorphic form
  Automorphic form
  In mathematics, the general notion of automorphic form is the extension to analytic functions, perhaps of several complex variables, of the theory of modular forms...
  
  
• Selberg trace formula
  Selberg trace formula
  In mathematics, the Selberg trace formula, introduced by , is an expression for the character of the unitary representation of G on the space L2 of square-integrable functions, where G is a Lie group and Γ a cofinite discrete group...
  
  
• Artin conjecture
• Sato–Tate conjecture
• Langlands program
  Langlands program
  The Langlands program is a web of far-reaching and influential conjectures that relate Galois groups in algebraic number theory to automorphic forms and representation theory of algebraic groups over local fields and adeles. It was proposed by ....
  
  
• modularity theorem

Diophantine equation

Diophantine equation

In mathematics, a Diophantine equation is an indeterminate polynomial equation that allows the variables to be integers only. Diophantine problems have fewer equations than unknown variables and involve finding integers that work correctly for all equations...

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• Pythagorean triple
  Pythagorean triple
  A Pythagorean triple consists of three positive integers a, b, and c, such that . Such a triple is commonly written , and a well-known example is . If is a Pythagorean triple, then so is for any positive integer k. A primitive Pythagorean triple is one in which a, b and c are pairwise coprime...
  
  
• Pell's equation
  Pell's equation
  Pell's equation is any Diophantine equation of the formx^2-ny^2=1\,where n is a nonsquare integer. The word Diophantine means that integer values of x and y are sought. Trivially, x = 1 and y = 0 always solve this equation...
  
  
• Elliptic curve
  Elliptic curve
  In mathematics, an elliptic curve is a smooth, projective algebraic curve of genus one, on which there is a specified point O. An elliptic curve is in fact an abelian variety — that is, it has a multiplication defined algebraically with respect to which it is a group — and O serves as the identity...
  
  
  • Nagell–Lutz theorem
    Nagell–Lutz theorem
    In mathematics, the Nagell–Lutz theorem is a result in the diophantine geometry of elliptic curves, which describes rational torsion points on elliptic curves over the integers.-Definition of the terms:Suppose that the equationy^2 = x^3 + ax^2 + bx + c \...
    
    
  • Mordell–Weil theorem
    Mordell–Weil theorem
    In mathematics, the Mordell–Weil theorem states that for an abelian variety A over a number field K, the group A of K-rational points of A is a finitely-generated abelian group, called the Mordell-Weil group...
    
    
  • Mazur's torsion theorem
    Mazur's torsion theorem
    In algebraic geometry Mazur's torsion theorem, due to Barry Mazur, classifies the possible torsion subgroups of the group of rational points on an elliptic curve defined over the rational numbers....
    
    
  • Congruent number
    Congruent number
    In mathematics, a congruent number is a positive integer that is the area of a right triangle with three rational number sides. A more general definition includes all positive rational numbers with this property....
    
    
  • Arithmetic of abelian varieties
    Arithmetic of abelian varieties
    In mathematics, the arithmetic of abelian varieties is the study of the number theory of an abelian variety, or family of those. It goes back to the studies of Fermat on what are now recognised as elliptic curves; and has become a very substantial area both in terms of results and conjectures...
    
    
  • Elliptic divisibility sequence
    Elliptic divisibility sequence
    In mathematics, an elliptic divisibility sequence is a sequence of integers satisfying a nonlinear recursion relation arising from division polynomials on elliptic curves. EDS were first defined, and their arithmetic properties studied, by Morgan Ward...
    
    s
• Fermat's Last Theorem
  Fermat's Last Theorem
  In number theory, Fermat's Last Theorem states that no three positive integers a, b, and c can satisfy the equation an + bn = cn for any integer value of n greater than two....
  
  
• Mordell conjecture
• Euler's sum of powers conjecture
• abc Conjecture
  Abc conjecture
  The abc conjecture is a conjecture in number theory, first proposed by Joseph Oesterlé and David Masser in 1985. The conjecture is stated in terms of three positive integers, a, b and c , which have no common factor and satisfy a + b = c...
  
  
• Catalan's conjecture
• Pillai's conjecture
• Hasse principle
  Hasse principle
  In mathematics, Helmut Hasse's local-global principle, also known as the Hasse principle, is the idea that one can find an integer solution to an equation by using the Chinese remainder theorem to piece together solutions modulo powers of each different prime number...
  
  
• Diophantine set
  Diophantine set
  In mathematics, a Diophantine equation is an equation of the form P=0 where P is a polynomial with integer coefficients...
  
  
• Matiyasevich's theorem
• One thousand seven hundred and twenty nine

Diophantine approximation

Diophantine approximation

In number theory, the field of Diophantine approximation, named after Diophantus of Alexandria, deals with the approximation of real numbers by rational numbers....

• Davenport–Schmidt theorem
  Davenport–Schmidt theorem
  In mathematics, specifically the area of Diophantine approximation, the Davenport–Schmidt theorem tells us how well a certain kind of real number can be approximated by another kind. Specifically it tells us that we can get a good approximation to irrational numbers that are not quadratic by using...
  
  
• Irrational number
  Irrational number
  In mathematics, an irrational number is any real number that cannot be expressed as a ratio a/b, where a and b are integers, with b non-zero, and is therefore not a rational number....
  
  
  • Square root of two
  • Quadratic irrational
    Quadratic irrational
    In mathematics, a quadratic irrational is an irrational number that is the solution to some quadratic equation with rational coefficients...
    
    
  • Integer square root
    Integer square root
    In number theory, the integer square root of a positive integer n is the positive integer m which is the greatest integer less than or equal to the square root of n,...
    
    
  • Algebraic number
    Algebraic number
    In mathematics, an algebraic number is a number that is a root of a non-zero polynomial in one variable with rational coefficients. Numbers such as π that are not algebraic are said to be transcendental; almost all real numbers are transcendental...
    
    
    • Pisot–Vijayaraghavan number
    • Salem number
  • Transcendental number
    Transcendental number
    In mathematics, a transcendental number is a number that is not algebraic—that is, it is not a root of a non-constant polynomial equation with rational coefficients. The most prominent examples of transcendental numbers are π and e...
    
    
    • e (mathematical constant)
      E (mathematical constant)
      The mathematical constant ' is the unique real number such that the value of the derivative of the function at the point is equal to 1. The function so defined is called the exponential function, and its inverse is the natural logarithm, or logarithm to base...
      
      
    • pi
      Pi
      ' is a mathematical constant that is the ratio of any circle's circumference to its diameter. is approximately equal to 3.14. Many formulae in mathematics, science, and engineering involve , which makes it one of the most important mathematical constants...
      
      , list of topics related to pi
    • Squaring the circle
      Squaring the circle
      Squaring the circle is a problem proposed by ancient geometers. It is the challenge of constructing a square with the same area as a given circle by using only a finite number of steps with compass and straightedge...
      
      
    • Proof that e is irrational
      Proof that e is irrational
      In mathematics, the series representation of Euler's number ecan be used to prove that e is irrational. Of the many representations of e, this is the Taylor series for the exponential function evaluated at y = 1.-Summary of the proof:...
      
      
    • Lindemann–Weierstrass theorem
      Lindemann–Weierstrass theorem
      In mathematics, the Lindemann–Weierstrass theorem is a result that is very useful in establishing the transcendence of numbers. It states that if 1, ...,  are algebraic numbers which are linearly independent over the rational numbers ', then 1, ...,  are algebraically...
      
      
    • Hilbert's seventh problem
      Hilbert's seventh problem
      Hilbert's seventh problem is one of David Hilbert's list of open mathematical problems posed in 1900. It concerns the irrationality and transcendence of certain numbers...
      
      
    • Gelfond–Schneider theorem
      Gelfond–Schneider theorem
      In mathematics, the Gelfond–Schneider theorem establishes the transcendence of a large class of numbers. It was originally proved independently in 1934 by Aleksandr Gelfond and Theodor Schneider...
      
      
  • Erdős–Borwein constant
• Liouville number
• Continued fraction
  Continued fraction
  In mathematics, a continued fraction is an expression obtained through an iterative process of representing a number as the sum of its integer part and the reciprocal of another number, then writing this other number as the sum of its integer part and another reciprocal, and so on...
  
  
  • Mathematical constant (sorted by continued fraction representation)
  • Khinchin's constant
  • Lévy's constant
    Lévy's constant
    In mathematics Lévy's constant occurs in an expression for the asymptotic behaviour of the denominators of the convergents of continued fractions....
    
    
  • Lochs' theorem
    Lochs' theorem
    In number theory, Lochs' theorem is a theorem concerning the rate of convergence of the continued fraction expansion of a typical real number. The theorem was proved by Gustav Lochs in 1964....
    
    
  • Gauss–Kuzmin–Wirsing operator
  • Minkowski's question mark function
    Minkowski's question mark function
    In mathematics, the Minkowski question mark function, sometimes called the slippery devil's staircase and denoted by ?, is a function possessing various unusual fractal properties, defined by Hermann Minkowski in 1904...
    
    
  • Generalized continued fraction
    Generalized continued fraction
    In complex analysis, a branch of mathematics, a generalized continued fraction is a generalization of regular continued fractions in canonical form, in which the partial numerators and partial denominators can assume arbitrary real or complex values....
    
    
• Kronecker's theorem
  Kronecker's theorem
  In mathematics, Kronecker's theorem is either of two theorems named after Leopold Kronecker.- The existence of extension fields :This is a theorem stating that a polynomial in a field, p ∈ F[x], has a root in an extension field E \supset F.For example, a polynomial in the reals such...
  
  
• Thue–Siegel–Roth theorem
  Thue–Siegel–Roth theorem
  In mathematics, the Thue–Siegel–Roth theorem, also known simply as Roth's theorem, is a foundational result in diophantine approximation to algebraic numbers. It is of a qualitative type, stating that a given algebraic number α may not have too many rational number approximations, that are 'very...
  
  
• Prouhet–Thue–Morse constant
• Gelfond–Schneider constant
• Equidistribution mod 1
• Beatty's theorem
• Littlewood conjecture
• Discrepancy function
  Discrepancy function
  A discrepancy function is a mathematical function which describes how closely a structural model conforms to observed data. Larger values of the discrepancy function indicate a poor fit of the model to data. In general, the parameter estimates for a given model are chosen so as to make the...
  
  
  • Low-discrepancy sequence
    Low-discrepancy sequence
    In mathematics, a low-discrepancy sequence is a sequence with the property that for all values of N, its subsequence x1, ..., xN has a low discrepancy....
    
    
  • Illustration of a low-discrepancy sequence
  • Constructions of low-discrepancy sequences
  • Halton sequences
    Halton sequences
    In statistics, Halton sequences are sequences used to generate points in space for numerical methods such as Monte Carlo simulations. Although these sequences are deterministic they are of low discrepancy, that is, appear to be random for many purposes. They were first introduced in 1960 and are...
    
    
• Geometry of numbers
  Geometry of numbers
  In number theory, the geometry of numbers studies convex bodies and integer vectors in n-dimensional space. The geometry of numbers was initiated by ....
  
  
  • Minkowski's theorem
    Minkowski's theorem
    In mathematics, Minkowski's theorem is the statement that any convex set in Rn which is symmetric with respect to the origin and with volume greater than 2n d contains a non-zero lattice point...
    
    
  • Pick's theorem
    Pick's theorem
    Given a simple polygon constructed on a grid of equal-distanced points such that all the polygon's vertices are grid points, Pick's theorem provides a simple formula for calculating the area A of this polygon in terms of the number i of lattice points in the interior located in the polygon and the...
    
    
  • Mahler's compactness theorem
    Mahler's compactness theorem
    In mathematics, Mahler's compactness theorem, proved by , is a foundational result on lattices in Euclidean space, characterising sets of lattices that are 'bounded' in a certain definite sense. Looked at another way, it explains the ways in which a lattice could degenerate in a sequence of...
    
    
• Mahler measure
• Effective results in number theory
  Effective results in number theory
  For historical reasons and in order to have application to the solution of Diophantine equations, results in number theory have been scrutinised more than in other branches of mathematics to see if their content is effectively computable...
  
  
• Mahler's theorem
  Mahler's theorem
  In mathematics, Mahler's theorem, introduced by , expresses continuous p-adic functions in terms of polynomials.In any field, one has the following result. Let=f-f\,be the forward difference operator...
  
  

Named primes

• Chen prime
  Chen prime
  A prime number p is called a Chen prime if p + 2 is either a prime or a product of two primes. The even number 2p + 2 therefore satisfies Chen's theorem....
  
  
• Cullen prime
• Fermat prime
• Sophie Germain prime
  Sophie Germain prime
  In number theory, a prime number p is a Sophie Germain prime if 2p + 1 is also prime. For example, 23 is a Sophie Germain prime because it is a prime and 2 × 23 + 1 = 47, and 47 is also a prime number...
  
  , safe prime
  Safe prime
  A safe prime is a prime number of the form 2p + 1, where p is also a prime. The first few safe primes are...
  
  
• Mersenne prime
  Mersenne prime
  In mathematics, a Mersenne number, named after Marin Mersenne , is a positive integer that is one less than a power of two: M_p=2^p-1.\,...
  
  
  • New Mersenne conjecture
  • Great Internet Mersenne Prime Search
    Great Internet Mersenne Prime Search
    The Great Internet Mersenne Prime Search is a collaborative project of volunteers who use freely available computer software to search for Mersenne prime numbers. The project was founded by George Woltman, who also wrote the software Prime95 and MPrime for the project...
    
    
• Newman–Shanks–Williams prime
• Primorial prime
  Primorial prime
  In mathematics, primorial primes are prime numbers of the form pn# ± 1, where:The first few primorial primes are, the largest known primorial prime is 843301# - 1 with 365,851 digits, found in 2010 by the PrimeGrid project....
  
  
• Wagstaff prime
• Wall–Sun–Sun prime
• Wieferich prime
  Wieferich prime
  In number theory, a Wieferich prime is a prime number p such that p2 divides 2p − 1 − 1, therefore connecting these primes with Fermat's little theorem, which states that every odd prime p divides 2p − 1 − 1...
  
  
• Wilson prime
  Wilson prime
  A Wilson prime, named after English mathematician John Wilson, is a prime number p such that p2 divides ! + 1, where "!" denotes the factorial function; compare this with Wilson's theorem, which states that every prime p divides ! + 1.The only known Wilson primes are 5, 13, and...
  
  
• Wolstenholme prime
  Wolstenholme prime
  In number theory, a Wolstenholme prime is a special type of prime number satisfying a stronger version of Wolstenholme's theorem. Wolstenholme's theorem is a congruence relation satisfied by all prime numbers greater than 7...
  
  
• Woodall prime
• Prime pages
  Prime pages
  The Prime Pages 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" lists for primes of various forms...
  
  

Combinatorial number theory

• Covering system
• Small set (combinatorics)
• Erdős–Ginzburg–Ziv theorem
• Polynomial method
• Van der Waerden's theorem
• Szemerédi's theorem
  Szemerédi's theorem
  In number theory, Szemerédi's theorem is a result that was formerly the Erdős–Turán conjecture...
  
  
• Collatz conjecture
  Collatz conjecture
  The Collatz conjecture is a conjecture in mathematics named after Lothar Collatz, who first proposed it in 1937. The conjecture is also known as the 3n + 1 conjecture, the Ulam conjecture , Kakutani's problem , the Thwaites conjecture , Hasse's algorithm The Collatz conjecture is a...
  
  
• Gilbreath's conjecture
  Gilbreath's conjecture
  Gilbreath's conjecture is a hypothesis, or a conjecture, in number theory regarding the sequences generated by applying the forward difference operator to consecutive prime numbers and leaving the results unsigned, and then repeating this process on consecutive terms in the resulting sequence, and...
  
  
• Erdős–Graham conjecture
• Znám's problem
  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 problem is named after the Slovak mathematician Štefan Znám, who suggested it in 1972, although other...
  
  

Computational number theory

Computational number theory

In mathematics, computational number theory, also known as algorithmic number theory, is the study of algorithms for performing number theoretic computations...

• Algorithmic number theory
• Residue number system
  Residue number system
  A residue number system represents a large integer using a set of smaller integers, so that computation may be performed more efficiently...
  
  
• Cunningham project
  Cunningham project
  The Cunningham project aims to find factors of large numbers of the formb^n \pm 1for b = 2, 3, 5, 6, 7, 10, 11, 12 and large exponents n. The project is named after Allan Joseph Champneys Cunningham, who published the first version of the table together with Herbert J. Woodall in 1925...
  
  
• Quadratic residuosity problem
  Quadratic residuosity problem
  The quadratic residuosity problem in computational number theory is the question of distinguishing by calculating the quadratic residues modulo N, where N is a composite number...
  
  

Primality test

Primality test

A primality test is an algorithm for determining whether an input number is prime. Amongst other fields of mathematics, it is used for cryptography. Unlike integer factorization, primality tests do not generally give prime factors, only stating whether the input number is prime or not...

s

• Prime factorization algorithm
• Trial division
  Trial division
  Trial division is the most laborious but easiest to understand of the integer factorization algorithms. Its ease of implementation makes it a viable integer factorization option for devices with little available memory, such as graphing calculators....
  
  
• Sieve of Eratosthenes
  Sieve of Eratosthenes
  In mathematics, the sieve of Eratosthenes , one of a number of prime number sieves, is a simple, ancient algorithm for finding all prime numbers up to a specified integer....
  
  
• Probabilistic algorithm
• Fermat primality test
  Fermat primality test
  The Fermat primality test is a probabilistic test to determine if a number is probable prime.-Concept:Fermat's little theorem states that if p is prime and 1 \le a...
  
  
  • Pseudoprime
    Pseudoprime
    In number theory, the Fermat pseudoprimes make up the most important class of pseudoprimes that come from Fermat's little theorem.- Definition :...
    
    
  • Carmichael number
  • Euler pseudoprime
    Euler pseudoprime
    In arithmetic, an odd composite integer n is called an Euler pseudoprime to base a, if a and n are coprime, and....
    
    
  • Euler–Jacobi pseudoprime
  • Fibonacci pseudoprime
  • Probable prime
    Probable prime
    In number theory, a probable prime is an integer that satisfies a specific condition also satisfied by all prime numbers. Different types of probable primes have different specific conditions...
    
    
• Miller–Rabin primality test
• Lucas–Lehmer primality test
  Lucas–Lehmer test
  In computational number theory, the Lucas test is a primality test for a natural number n; it requires that the prime factors of n − 1 be already known. It is the basis of the Pratt certificate that gives a concise verification that n is prime....
  
  
• Lucas–Lehmer test for Mersenne numbers
  Lucas–Lehmer test for Mersenne numbers
  In mathematics, the Lucas–Lehmer test is a primality test for Mersenne numbers. The test was originally developed by Édouard Lucas in 1856, and subsequently improved by Lucas in 1878 and Derrick Henry Lehmer in the 1930s.-The test:...
  
  
• AKS primality test
  AKS primality test
  The AKS primality test is a deterministic primality-proving algorithm created and published by three Indian Institute of Technology Kanpur computer scientists, Manindra Agrawal, Neeraj Kayal, and Nitin Saxena, on August 6, 2002, in a paper titled "PRIMES is in P"...
  
  
• NewPGen
  NewPGen
  NewPGen is a program used by researchers looking for large prime numbers. It is a program that is used to rapidly presieve a set of candidate numbers, removing those that are definitely composite numbers...
  
  

Integer factorization

Integer factorization

In number theory, integer factorization or prime factorization is the decomposition of a composite number into smaller non-trivial divisors, which when multiplied together equal the original integer....

• Pollard's p − 1 algorithm
• Pollard's rho algorithm
  Pollard's rho algorithm
  Pollard's rho algorithm is a special-purpose integer factorization algorithm. It was invented by John Pollard in 1975. It is particularly effective at splitting composite numbers with small factors.-Core ideas:...
  
  
• Lenstra elliptic curve factorization
  Lenstra elliptic curve factorization
  The Lenstra elliptic curve factorization or the elliptic curve factorization method is a fast, sub-exponential running time algorithm for integer factorization which employs elliptic curves. For general purpose factoring, ECM is the third-fastest known factoring method...
  
  
• Quadratic sieve
  Quadratic sieve
  The quadratic sieve algorithm is a modern integer factorization algorithm and, in practice, the second fastest method known . It is still the fastest for integers under 100 decimal digits or so, and is considerably simpler than the number field sieve...
  
  
• Special number field sieve
  Special number field sieve
  In number theory, a branch of mathematics, the special number field sieve is a special-purpose integer factorization algorithm. The general number field sieve was derived from it....
  
  
• General number field sieve
  General number field sieve
  In number theory, the general number field sieve is the most efficient classical algorithm known for factoring integers larger than 100 digits...
  
  
• Shor's algorithm
  Shor's algorithm
  Shor's algorithm, named after mathematician Peter Shor, is a quantum algorithm for integer factorization formulated in 1994...
  
  
• RSA Factoring Challenge
  RSA Factoring Challenge
  The RSA Factoring Challenge was a challenge put forward by RSA Laboratories on March 18, 1991 to encourage research into computational number theory and the practical difficulty of factoring large integers and cracking RSA keys used in cryptography...
  
  
  • FAFNER
    FAFNER
    Factoring via Network-Enabled Recursion was a 1995 project trying to solve the RSA-130 factoring problem.It was an internet-based sieving effort from Cooperating Systems Corporation...
    
    

Pseudo-random numbers

• Pseudorandom number generator
  Pseudorandom number generator
  A pseudorandom number generator , also known as a deterministic random bit generator , is an algorithm for generating a sequence of numbers that approximates the properties of random numbers...
  
  
  • Pseudorandomness
    Pseudorandomness
    A pseudorandom process is a process that appears to be random but is not. Pseudorandom sequences typically exhibit statistical randomness while being generated by an entirely deterministic causal process...
    
    
  • Cryptographically secure pseudo-random number generator
• Middle-square method
  Middle-square method
  In mathematics, the middle-square method is a method of generating pseudorandom numbers. In practice it is not a good method, since its period is usually very short and it has some crippling weaknesses...
  
  
• Blum Blum Shub
• ISAAC
  ISAAC (cipher)
  ISAAC is a cryptographically secure pseudorandom number generator and a stream cipher designed by Robert J. Jenkins Jr. in 1996.- Operation :...
  
  
• Lagged Fibonacci generator
  Lagged Fibonacci generator
  A Lagged Fibonacci generator is an example of a pseudorandom number generator. This class of random number generator is aimed at being an improvement on the 'standard' linear congruential generator...
  
  
• Linear congruential generator
  Linear congruential generator
  A Linear Congruential Generator represents one of the oldest and best-known pseudorandom number generator algorithms. The theory behind them is easy to understand, and they are easily implemented and fast....
  
  
  • Mersenne twister
• Linear feedback shift register
  Linear feedback shift register
  A linear feedback shift register is a shift register whose input bit is a linear function of its previous state.The most commonly used linear function of single bits is XOR...
  
  
• Shrinking generator
  Shrinking generator
  In cryptography, the shrinking generator is a form of pseudorandom number generator intended to be used in a stream cipher. It was published in Crypto 1993 by Don Coppersmith, Hugo Krawczyk, and Yishay Mansour....
  
  
• Stream cipher
  Stream cipher
  In cryptography, a stream cipher is a symmetric key cipher where plaintext digits are combined with a pseudorandom cipher digit stream . In a stream cipher the plaintext digits are encrypted one at a time, and the transformation of successive digits varies during the encryption...
  
  

History

• Disquisitiones Arithmeticae
  Disquisitiones Arithmeticae
  The Disquisitiones Arithmeticae is a textbook of number theory written in Latin by Carl Friedrich Gauss in 1798 when Gauss was 21 and first published in 1801 when he was 24...
  
  
• On the Number of Primes Less Than a Given Magnitude
  On the Number of Primes Less Than a Given Magnitude
  die Anzahl der Primzahlen unter einer gegebenen is a seminal 8-page paper by Bernhard Riemann published in the November 1859 edition of the Monatsberichte der Königlich Preußischen Akademie der Wissenschaften zu Berlin.Although it is the only paper he ever published on number theory, it...
  
  
• Vorlesungen über Zahlentheorie
  Vorlesungen über Zahlentheorie
  ' is a textbook of number theory written by German mathematicians Lejeune Dirichlet and Richard Dedekind, and published in 1863....
  
  
• Prime Obsession
  Prime Obsession
  Prime Obsession: Bernhard Riemann and the Greatest Unsolved Problem in Mathematics is a historical book on mathematics by John Derbyshire, detailing the history of the Riemann hypothesis, named for Bernhard Riemann, and some of its applications...