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 arithmetic function, integer-valued functions. German mathematician Carl Friedrich Gauss (1777â ...
topics, by Wikipedia page. See also:
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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 problems posed purely for their own sake.
See list o ...
Composite number
A composite number is a positive integer that can be formed by multiplying two smaller positive integers. Equivalently, it is a positive integer that has at least one divisor other than 1 and itself. Every positive integer is composite, prime, ...
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Highly composite number
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A highly composite number is a positive integer with more divisors than any smaller positive integer has. The related concept of largely composite number refers to a positive integer which has at least as many divisors as any smaller ...
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Even and odd numbers
In mathematics, parity is the property of an integer of whether it is even or odd. An integer is even if it is a multiple of two, and odd if it is not.. For example, â4, 0, 82 are even because
\begin
-2 \cdot 2 &= -4 \\
0 \cdot 2 &= 0 \\
41 ...
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Parity
Parity may refer to:
* Parity (computing)
** Parity bit in computing, sets the parity of data for the purpose of error detection
** Parity flag in computing, indicates if the number of set bits is odd or even in the binary representation of the r ...
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Divisor
In mathematics, a divisor of an integer n, also called a factor of n, is an integer m that may be multiplied by some integer to produce n. In this case, one also says that n is a multiple of m. An integer n is divisible or evenly divisible by ...
,
aliquot part
In mathematics, a divisor of an integer n, also called a factor of n, is an integer m that may be multiplied by some integer to produce n. In this case, one also says that n is a multiple of m. An integer n is divisible or evenly divisible by ...
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 that is divisible by bot ...
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Euclidean algorithm
In mathematics, the Euclidean algorithm,Some widely used textbooks, such as I. N. Herstein's ''Topics in Algebra'' and Serge Lang's ''Algebra'', use the term "Euclidean algorithm" to refer to Euclidean division or Euclid's algorithm, is an effi ...
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'', for which ''n''/''m'' is again an integer (which is necessarily also a divisor of ''n''). For example, 3 is a divisor of 21, sin ...
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Prime number
A prime number (or a prime) is a natural number greater than 1 that is not a product of two smaller natural numbers. A natural number greater than 1 that is not prime is called a composite number. For example, 5 is prime because the only ways ...
Prime factor
A prime number (or a prime) is a natural number greater than 1 that is not a product of two smaller natural numbers. A natural number greater than 1 that is not prime is called a composite number. For example, 5 is prime because the only ways ...
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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. It has no prime factors and is ne ...
Square number
In mathematics, a square number or perfect square is an integer that is the square (algebra), square of an integer; in other words, it is the multiplication, product of some integer with itself. For example, 9 is a square number, since it equals ...
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Power of two
A power of two is a number of the form where is an integer, that is, the result of exponentiation with number two as the base and integer as the exponent.
In a context where only integers are considered, is restricted to non-negative ...
Fraction
A fraction (from la, fractus, "broken") represents a part of a whole or, more generally, any number of equal parts. When spoken in everyday English, a fraction describes how many parts of a certain size there are, for example, one-half, eight ...
s
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Rational number
In mathematics, a rational number is a number that can be expressed as the quotient or fraction of two integers, a numerator and a non-zero denominator . For example, is a rational number, as is every integer (e.g. ). The set of all ration ...
in lowest terms
An irreducible fraction (or fraction in lowest terms, simplest form or reduced fraction) is a fraction (mathematics), fraction in which the numerator and denominator are integers that have no other common divisors than 1 (and â1, when negative nu ...
Ford circle
In mathematics, a Ford circle is a circle with center at (p/q,1/(2q^2)) and radius 1/(2q^2), where p/q is an irreducible fraction, i.e. p and q are coprime integers. Each Ford circle is tangent to the horizontal axis y=0, and any two Ford circles ...
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 of the Dedekind eta function. They have sub ...
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Egyptian fraction
An Egyptian fraction is a finite sum of distinct unit fractions, such as
\frac+\frac+\frac.
That is, each fraction in the expression has a numerator equal to 1 and a denominator that is a positive integer, and all the denominators differ from each ...
Modular arithmetic
In mathematics, modular arithmetic is a system of arithmetic for integers, where numbers "wrap around" when reaching a certain value, called the modulus. The modern approach to modular arithmetic was developed by Carl Friedrich Gauss in his book ...
Linear congruence theorem
In mathematics, the Chinese remainder theorem states that if one knows the remainders of the Euclidean division of an integer ''n'' by several integers, then one can determine uniquely the remainder of the division of ''n'' by the product of thes ...
Chinese remainder theorem
In mathematics, the Chinese remainder theorem states that if one knows the remainders of the Euclidean division of an integer ''n'' by several integers, then one can determine uniquely the remainder of the division of ''n'' by the product of thes ...
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Fermat's little theorem
Fermat's little theorem states that if ''p'' is a prime number, then for any integer ''a'', the number a^p - a is an integer multiple of ''p''. In the notation of modular arithmetic, this is expressed as
: a^p \equiv a \pmod p.
For example, if = ...
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Proofs of Fermat's little theorem This article collects together a variety of proofs of Fermat's little theorem, which states that
:a^p \equiv a \pmod p
for every prime number ''p'' and every integer ''a'' (see modular arithmetic).
Simplifications
Some of the proofs of Fermat's lit ...
Euler's totient function
In number theory, Euler's totient function counts the positive integers up to a given integer that are relatively prime to . It is written using the Greek letter phi as \varphi(n) or \phi(n), and may also be called Euler's phi function. In ot ...
Euler's theorem
In number theory, Euler's theorem (also known as the FermatâEuler theorem or Euler's totient theorem) states that, if and are coprime positive integers, and \varphi(n) is Euler's totient function, then raised to the power \varphi(n) is congru ...
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Wilson's theorem
In algebra and number theory, Wilson's theorem states that a natural number ''n'' > 1 is a prime number if and only if the product of all the positive integers less than ''n'' is one less than a multiple of ''n''. That is (using the notations of m ...
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Primitive root modulo n
In modular arithmetic, a number is a primitive root modulo if every number coprime to is congruent to a power of modulo . That is, is a ''primitive root modulo'' if for every integer coprime to , there is some integer for which ...
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Multiplicative order In number theory, given a positive integer ''n'' and an integer ''a'' coprime to ''n'', the multiplicative order of ''a'' modulo ''n'' is the smallest positive integer ''k'' such that a^k\ \equiv\ 1 \pmod n.
In other words, the multiplicative order ...
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Discrete logarithm
In mathematics, for given real numbers ''a'' and ''b'', the logarithm log''b'' ''a'' is a number ''x'' such that . Analogously, in any group ''G'', powers ''b'k'' can be defined for all integers ''k'', and the discrete logarithm log''b' ...
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Quadratic residue
In number theory, an integer ''q'' is called a quadratic residue modulo ''n'' if it is congruent to a perfect square modulo ''n''; i.e., if there exists an integer ''x'' such that:
:x^2\equiv q \pmod.
Otherwise, ''q'' is called a quadratic no ...
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Euler's criterion In number theory, Euler's criterion is a formula for determining whether an integer is a quadratic residue modulo a prime. Precisely,
Let ''p'' be an odd prime and ''a'' be an integer coprime to ''p''. Then
:
a^ \equiv
\begin
\;\;\,1\pmod& \text ...
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Legendre symbol
In number theory, the Legendre symbol is a multiplicative function with values 1, â1, 0 that is a quadratic character modulo an odd prime number ''p'': its value at a (nonzero) quadratic residue mod ''p'' is 1 and at a non-quadratic residu ...
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'' and ''y'' satisfying the equali ...
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Luhn formula
The Luhn algorithm or Luhn formula, also known as the " modulus 10" or "mod 10" algorithm, named after its creator, IBM scientist Hans Peter Luhn, is a simple checksum formula used to validate a variety of identification numbers, such as credit ca ...
ErdĆsâKac theorem In number theory, the ErdĆsâKac theorem, named after Paul ErdĆs and Mark Kac, and also known as the fundamental theorem of probabilistic number theory, states that if ''Ï''(''n'') is the number of distinct prime factors of ''n'', then, loosel ...
Integer partition
In number theory and combinatorics, a partition of a positive integer , also called an integer partition, is a way of writing as a sum of positive integers. Two sums that differ only in the order of their summands are considered the same part ...
Landau's function In mathematics, Landau's function ''g''(''n''), named after Edmund Landau, is defined for every natural number ''n'' to be the largest order of an element of the symmetric group ''S'n''. Equivalently, ''g''(''n'') is the largest least common mu ...
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Pentagonal number theorem
In mathematics, the pentagonal number theorem, originally due to Euler, relates the product and series representations of the Euler function. It states that
:\prod_^\left(1-x^\right)=\sum_^\left(-1\right)^x^=1+\sum_^\infty(-1)^k\left(x^+x^\right ...
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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.
Given an arithmetic function f and a prime p, define the formal power seri ...
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Lambert series
In mathematics, a Lambert series, named for Johann Heinrich Lambert, is a series taking the form
:S(q)=\sum_^\infty a_n \frac .
It can be resumed formally by expanding the denominator:
:S(q)=\sum_^\infty a_n \sum_^\infty q^ = \sum_^\infty b_m ...
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 Peter Gustav Lejeune Dirichlet's 1837 introduction of Diric ...
: additive problems
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Twin prime
A twin prime is a prime number that is either 2 less or 2 more than another prime numberâfor example, either member of the twin prime pair (41, 43). In other words, a twin prime is a prime that has a prime gap of two. Sometimes the term ''twin pr ...
Cousin prime
In number theory, 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.
The cousin primes (sequences and in OE ...
Sexy prime
In number theory, sexy primes are prime numbers that differ from each other by 6. For example, the numbers 5 and 11 are both sexy primes, because both are prime and .
The term "sexy prime" is a pun stemming from the Latin word for six: .
If o ...
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.
Definition
A Cunningham chain of the first kin ...
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Goldbach's conjecture
Goldbach's conjecture is one of the oldest and best-known unsolved problems in number theory and all of mathematics. It states that every even natural number greater than 2 is the sum of two prime numbers.
The conjecture has been shown to hold ...
Second HardyâLittlewood conjecture
In number theory, the second HardyâLittlewood conjecture concerns the number of primes in intervals. Along with the first HardyâLittlewood conjecture, the second HardyâLittlewood conjecture was proposed by G. H. Hardy and John Edensor Little ...
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HardyâLittlewood circle method
In mathematics, the HardyâLittlewood circle method is a technique of analytic number theory. It is named for G. H. Hardy and J. E. Littlewood, who developed it in a series of papers on Waring's problem.
History
The initial idea is usually at ...
BatemanâHorn conjecture In number theory, the BatemanâHorn conjecture is a statement concerning the frequency of prime numbers among the values of a system of polynomials, named after mathematicians Paul T. Bateman and Roger A. Horn who proposed it in 1962. It provides ...
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Waring's problem
In number theory, Waring's problem asks whether each natural number ''k'' has an associated positive integer ''s'' such that every natural number is the sum of at most ''s'' natural numbers raised to the power ''k''. For example, every natural num ...
Euler's four-square identity
In mathematics, Euler's four-square identity says that the product of two numbers, each of which is a sum of four square (algebra), squares, is itself a sum of four squares.
Algebraic identity
For any pair of quadruples from a commutative ring, th ...
Taxicab number
In mathematics, the ''n''th taxicab number, typically denoted Ta(''n'') or Taxicab(''n''), also called the ''n''th HardyâRamanujan number, is defined as the smallest integer that can be expressed as a sum of two ''positive'' integer cubes in ...
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Generalized taxicab number
In mathematics, the generalized taxicab number ''Taxicab''(''k'', ''j'', ''n'') is the smallest number â if it exists â that can be expressed as the sum of ''j'' ''k''th positive powers in ''n'' different ways. For ''k'' = 3 and ''j'' = 2, the ...
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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 numbers exist for all ''n'', whi ...
LandauâRamanujan constant In mathematics and the field of number theory, the LandauâRamanujan constant is the positive real number ''b'' that occurs in a theorem proved by Edmund Landau in 1908, stating that for large x, the number of positive integers below x that are the ...
Seventeen or Bust
Seventeen or Bust was a volunteer computing project started in March 2002 to solve the last seventeen cases in the Sierpinski problem. The project solved eleven cases before a server loss in April 2016 forced it to cease operations. Work on the ...
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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 ''H''(''n'') = the largest exponent a ...
Algebraic number theory
Algebraic number theory is a branch of number theory that uses the techniques of abstract algebra to study the integers, rational numbers, and their generalizations. Number-theoretic questions are expressed in terms of properties of algebraic ob ...
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list of algebraic number theory topics
This is a list of algebraic number theory topics.
Basic topics
These topics are basic to the field, either as prototypical examples, or as basic objects of study.
*Algebraic number field
**Gaussian integer, Gaussian rational
**Quadratic field
**C ...
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Quadratic form
In mathematics, a quadratic form is a polynomial with terms all of degree two ("form" is another name for a homogeneous polynomial). For example,
:4x^2 + 2xy - 3y^2
is a quadratic form in the variables and . The coefficients usually belong to a ...
s
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Unimodular lattice
In geometry and mathematical group theory, a unimodular lattice is an integral lattice of determinant 1 or −1. For a lattice in ''n''-dimensional Euclidean space, this is equivalent to requiring that the volume of any fundamen ...
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Fermat's theorem on sums of two squares
In additive number theory, Fermat's theorem on sums of two squares states that an odd prime ''p'' can be expressed as:
:p = x^2 + y^2,
with ''x'' and ''y'' integers, if and only if
:p \equiv 1 \pmod.
The prime numbers for which this is true ar ...
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Proofs of Fermat's theorem on sums of two squares
In additive number theory, Fermat's theorem on sums of two squares states that an odd prime ''p'' can be expressed as:
:p = x^2 + y^2,
with ''x'' and ''y'' integers, if and only if
:p \equiv 1 \pmod.
The prime numbers for which this is true ar ...
L-function
In mathematics, an ''L''-function is a meromorphic function on the complex plane, associated to one out of several categories of mathematical objects. An ''L''-series is a Dirichlet series, usually convergent on a half-plane, that may give ris ...
s
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Riemann zeta function
The Riemann zeta function or EulerâRiemann zeta function, denoted by the Greek letter (zeta), is a mathematical function of a complex variable defined as \zeta(s) = \sum_^\infty \frac = \frac + \frac + \frac + \cdots for \operatorname(s) > ...
Von StaudtâClausen theorem
In number theory, the von StaudtâClausen theorem is a result determining the fractional part of Bernoulli numbers, found independently by
and .
Specifically, if ''n'' is a positive integer and we add 1/''p'' to the Bernoulli number ''B''2''n'' ...
Prime number theorem
In mathematics, the prime number theorem (PNT) describes the asymptotic distribution of the prime numbers among the positive integers. It formalizes the intuitive idea that primes become less common as they become larger by precisely quantifying ...
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Prime-counting function
In mathematics, the prime-counting function is the function counting the number of prime numbers less than or equal to some real number ''x''. It is denoted by (''x'') (unrelated to the number ).
History
Of great interest in number theory is t ...
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MeisselâLehmer algorithm The MeisselâLehmer algorithm (after Ernst Meissel and Derrick Henry Lehmer) is an algorithm that computes exact values of the prime-counting function.
Description
The problem of counting the exact number of primes less than or equal to x, wit ...
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Offset logarithmic integral
In mathematics, the logarithmic integral function or integral logarithm li(''x'') is a special function. It is relevant in problems of physics and has number theoretic significance. In particular, according to the prime number theorem, it is a ...
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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 \pi(x). Its value is now known to be 1.
Examination of available n ...
Bertrand's postulate
In number theory, Bertrand's postulate is a theorem stating that for any integer n > 3, there always exists at least one prime number p with
:n < p < 2n - 2.
A less restrictive formulation is: for every , there is always ...
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Proof of Bertrand's postulate In mathematics, Bertrand's postulate (actually a theorem) states that for each n \ge 2 there is a prime p such that n. It was first
Riemann hypothesis
In mathematics, the Riemann hypothesis is the conjecture that the Riemann zeta function has its zeros only at the negative even integers and complex numbers with real part . Many consider it to be the most important unsolved problem in ...
HilbertâPĂłlya conjecture In mathematics, the HilbertâPĂłlya conjecture states that the non-trivial zeros of the Riemann zeta function correspond to eigenvalues of a self-adjoint operator. It is a possible approach to the Riemann hypothesis, by means of spectral theor ...
Mertens conjecture
In mathematics, the Mertens conjecture is the statement that the Mertens function M(n) is bounded by \pm\sqrt. Although now disproven, it had been shown to imply the Riemann hypothesis. It was conjectured by Thomas Joannes Stieltjes, in an 1885 ...
De BruijnâNewman constant
The de BruijnâNewman constant, denoted by Î and named after Nicolaas Govert de Bruijn and Charles M. Newman, is a mathematical constant defined via the zero of a function, zeros of a certain function (mathematics), function ''H''(''λ'', ...
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'' is als ...
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Linnik's theorem Linnik's theorem in analytic number theory answers a natural question after Dirichlet's theorem on arithmetic progressions. It asserts that there exist positive ''c'' and ''L'' such that, if we denote p(''a'',''d'') the least primes in arithmetic pr ...
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ElliottâHalberstam conjecture
In number theory, the ElliottâHalberstam conjecture is a conjecture about the distribution of prime numbers in arithmetic progressions. It has many applications in sieve theory. It is named for Peter D. T. A. Elliott and Heini Halberstam, who st ...
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 \mathbb of rational numbers. Generally speaking, a prime integer will factor into several ideal ...
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Local zeta function In number theory, the local zeta function (sometimes called the congruent zeta function or the HasseâWeil zeta function) is defined as
:Z(V, s) = \exp\left(\sum_^\infty \frac (q^)^m\right)
where is a non-singular -dimensional projective algebr ...
Modular form
In mathematics, a modular form is a (complex) analytic function on the upper half-plane satisfying a certain kind of functional equation with respect to the Group action (mathematics), group action of the modular group, and also satisfying a grow ...
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modular group
In mathematics, the modular group is the projective special linear group of matrices with integer coefficients and determinant 1. The matrices and are identified. The modular group acts on the upper-half of the complex plane by fractional l ...
RamanujanâPetersson conjecture In mathematics, the Ramanujan conjecture, due to , states that Ramanujan's tau function given by the Fourier coefficients of the cusp form of weight
:\Delta(z)= \sum_\tau(n)q^n=q\prod_\left (1-q^n \right)^ = q-24q^2+252q^3- 1472q^4 + 4830q^5-\ ...
Automorphic form
In harmonic analysis and number theory, an automorphic form is a well-behaved function from a topological group ''G'' to the complex numbers (or complex vector space) which is invariant under the action of a discrete subgroup \Gamma \subset G of ...
Sato–Tate conjecture
In mathematics, the SatoâTate conjecture is a statistical statement about the family of elliptic curves ''Ep'' obtained from an elliptic curve ''E'' over the rational numbers by reduction modulo almost all prime numbers ''p''. Mikio Sato and Jo ...
Diophantine equation
In mathematics, a Diophantine equation is an equation, typically a polynomial equation in two or more unknowns with integer coefficients, such that the only solutions of interest are the integer ones. A linear Diophantine equation equates to a c ...
Elliptic curve
In mathematics, an elliptic curve is a smooth, projective, algebraic curve of genus one, on which there is a specified point . An elliptic curve is defined over a field and describes points in , the Cartesian product of with itself. If ...
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NagellâLutz theorem
In mathematics, the NagellâLutz theorem is a result in the diophantine equation, diophantine geometry of elliptic curves, which describes rational number, rational Torsion (algebra), torsion points on elliptic curves over the integers.
It is name ...
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MordellâWeil theorem
In mathematics, the MordellâWeil theorem states that for an abelian variety A over a number field K, the group A(K) of ''K''-rational points of A is a finitely-generated abelian group, called the MordellâWeil group. The case with A an elli ...
Congruent number
In number theory, 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.
The sequence of (integer) cong ...
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, and their arithmetic properties studied, by M ...
s
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Mordell curve
In algebra, a Mordell curve is an elliptic curve of the form ''y''2 = ''x''3 + ''n'', where ''n'' is a fixed non-zero integer.
These curves were closely studied by Louis Mordell, from the point of view of determining their integer points. He sho ...
Mordell conjecture
Louis Joel Mordell (28 January 1888 â 12 March 1972) was an American-born British mathematician, known for pioneering research in number theory. He was born in Philadelphia, United States, in a Jewish family of Lithuanian extraction.
Educati ...
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Euler's sum of powers conjecture
Euler's conjecture is a disproved conjecture in mathematics related to Fermat's Last Theorem. It was proposed by Leonhard Euler in 1769. It states that for all integers and greater than 1, if the sum of many th powers of positive integers is ...
Pillai's conjecture
Catalan's conjecture (or MihÄilescu's theorem) is a theorem in number theory that was conjectured by the mathematician EugĂšne Charles Catalan in 1844 and proven in 2002 by Preda MihÄilescu at Paderborn University. The integers 23 and 32 are ...
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Hasse principle In mathematics, Helmut Hasse's localâglobal 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 eac ...
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Diophantine set
In mathematics, a Diophantine equation is an equation of the form ''P''(''x''1, ..., ''x'j'', ''y''1, ..., ''y'k'') = 0 (usually abbreviated ''P''(', ') = 0) where ''P''(', ') is a polynomial with integer coefficients, where ''x''1, ..., '' ...
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Matiyasevich's theorem
In mathematics, a Diophantine equation is an equation of the form ''P''(''x''1, ..., ''x'j'', ''y''1, ..., ''y'k'') = 0 (usually abbreviated ''P''(', ') = 0) where ''P''(', ') is a polynomial with integer coefficients, where ''x''1, ..., '' ...
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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 problems written by Zhang Qiujian. It is one of ...
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1729
Events
January–March
* January 8 – Frederick, the eldest son of King George II of Great Britain is made Prince of Wales at the age of 21, a few months after he comes to Britain for the first time after growing up in Hanover ...
DavenportâSchmidt theorem
In mathematics, specifically the area of Diophantine approximation, the DavenportâSchmidt 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 ...
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'',
: \mbox( n ) = \lfloor \sqrt n \rfloor.
For example ...
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Algebraic number
An algebraic number is a number that is a root of a non-zero polynomial in one variable with integer (or, equivalently, rational) coefficients. For example, the golden ratio, (1 + \sqrt)/2, is an algebraic number, because it is a root of the po ...
Transcendental number
In mathematics, a transcendental number is a number that is not algebraicâthat is, not the root of a non-zero polynomial of finite degree with rational coefficients. The best known transcendental numbers are and .
Though only a few classes ...
list of topics related to pi
A ''list'' is any set of items in a row. List or lists may also refer to:
People
* List (surname)
Organizations
* List College, an undergraduate division of the Jewish Theological Seminary of America
* SC Germania List, German rugby unio ...
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Squaring the circle
Squaring the circle is a problem in geometry first proposed in Greek mathematics. It is the challenge of constructing a square with the area of a circle by using only a finite number of steps with a compass and straightedge. The difficulty ...
GelfondâSchneider theorem
In mathematics, the GelfondâSchneider theorem establishes the transcendence of a large class of numbers.
History
It was originally proved independently in 1934 by Aleksandr Gelfond and Theodor Schneider.
Statement
: If ''a'' and ''b'' are ...
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ErdĆsâBorwein constant The ErdĆsâBorwein constant is the sum of the Reciprocal (mathematics), reciprocals of the Mersenne prime, Mersenne numbers. It is named after Paul ErdĆs and Peter Borwein.
By definition it is:
:E=\sum_^\frac\approx1.606695152415291763\dots
Eq ...
Khinchin's constant In number theory, Aleksandr Yakovlevich Khinchin proved that for almost all real numbers ''x'', coefficients ''a'i'' of the continued fraction expansion of ''x'' have a finite geometric mean that is independent of the value of ''x'' and is kno ...
Lochs' theorem
In number theory, Lochs's theorem concerns the rate of convergence of the continued fraction expansion of a typical real number. A proof of the theorem was published in 1964 by Gustav Lochs.
The theorem states that for almost all real numbers in ...
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GaussâKuzminâWirsing operator In mathematics, the GaussâKuzminâWirsing operator is the transfer operator of the Gauss map that takes a positive number to the fractional part of its reciprocal. (This is not the same as the Gauss map in differential geometry.) It is named af ...
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Minkowski's question mark function
In mathematics, the Minkowski question-mark function, denoted , is a function with unusual fractal properties, defined by Hermann Minkowski in 1904. It maps quadratic irrational numbers to rational numbers on the unit interval, via an expressio ...
GelfondâSchneider constant
The GelfondâSchneider constant or Hilbert number is two to the power of the square root of two:
:2 = ...
which was proved to be a transcendental number by Rodion Kuzmin in 1930.
In 1934, Aleksandr Gelfond and Theodor Schneider independently prov ...
Beatty's theorem
In mathematics, a Beatty sequence (or homogeneous Beatty sequence) is the sequence of integers found by taking the Floor and ceiling functions, floor of the positive Multiple (mathematics), multiples of a positive irrational number. Beatty sequence ...
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Littlewood conjecture
In mathematics, the Littlewood conjecture is an open problem () in Diophantine approximation, proposed by John Edensor Littlewood around 1930. It states that for any two real numbers α and ÎČ,
:\liminf_ \ n\,\Vert n\alpha\Vert \,\Vert n\beta\Ver ...
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Discrepancy function In structural equation modeling, a discrepancy function is a mathematical function which describes how closely a structural model conforms to observed data; it is a measure of goodness of fit. Larger values of the discrepancy function indicate a poo ...
Constructions of low-discrepancy sequences In mathematics, a low-discrepancy sequence is a sequence with the property that for all values of ''N'', its subsequence ''x''1, ..., ''x'N'' has a low discrepancy.
Roughly speaking, the discrepancy of a sequence is low if the proportion of poi ...
Geometry of numbers Geometry of numbers is the part of number theory which uses geometry for the study of algebraic numbers. Typically, a ring of algebraic integers is viewed as a lattice in \mathbb R^n, and the study of these lattices provides fundamental information ...
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Minkowski's theorem
In mathematics, Minkowski's theorem is the statement that every convex set in \mathbb^n which is symmetric with respect to the origin and which has volume greater than 2^n contains a non-zero integer point (meaning a point in \Z^n that is not t ...
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Pick's theorem
In geometry, Pick's theorem provides a formula for the area of a simple polygon with integer vertex coordinates, in terms of the number of integer points within it and on its boundary. The result was first described by Georg Alexander Pick in 18 ...
Mahler's theorem
In mathematics, Mahler's theorem, introduced by , expresses continuous ''p''-adic functions in terms of polynomials. Over any field of characteristic 0, one has the following result:
Let (\Delta f)(x)=f(x+1)-f(x) be the forward difference operat ...
Brun sieve In the field of number theory, the Brun sieve (also called Brun's pure sieve) is a technique for estimating the size of "sifted sets" of positive integers which satisfy a set of conditions which are expressed by congruences. It was developed by V ...
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Function field sieve In mathematics the Function Field Sieve is one of the most efficient algorithms to solve the Discrete Logarithm Problem (DLP) in a finite field. It has heuristic subexponential complexity. Leonard Adleman developed it in 1994 and then elaborated i ...
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General number field sieve
In number theory, the general number field sieve (GNFS) is the most efficient classical algorithm known for factoring integers larger than . Heuristically, its complexity for factoring an integer (consisting of bits) is of the form
:\exp\left( ...
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Large sieve
The large sieve is a method (or family of methods and related ideas) in analytic number theory. It is a type of sieve where up to half of all residue classes of numbers are removed, as opposed to small sieves such as the Selberg sieve wherein only ...
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Larger sieve In number theory, the larger sieve is a sieve invented by Patrick X. Gallagher. The name denotes a heightening of the large sieve. Combinatorial sieves like the Selberg sieve are strongest, when only a few residue classes are removed, while the te ...
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Quadratic sieve The quadratic sieve algorithm (QS) is an integer factorization algorithm and, in practice, the second fastest method known (after the general number field sieve). It is still the fastest for integers under 100 decimal digits or so, and is considerab ...
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Selberg sieve
In number theory, the Selberg sieve is a technique for estimating the size of "sifted sets" of positive integers which satisfy a set of conditions which are expressed by congruences. It was developed by Atle Selberg in the 1940s.
Description
In ...
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Sieve of Atkin In mathematics, the sieve of Atkin is a modern algorithm for finding all prime numbers up to a specified integer. Compared with the ancient sieve of Eratosthenes, which marks off multiples of primes, the sieve of Atkin does some preliminary work a ...
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Sieve of Eratosthenes
In mathematics, the sieve of Eratosthenes is an ancient algorithm for finding all prime numbers up to any given limit.
It does so by iteratively marking as composite (i.e., not prime) the multiples of each prime, starting with the first prime n ...
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Sieve of Sundaram In mathematics, the sieve of Sundaram is a variant of the sieve of Eratosthenes, a simple deterministic algorithm for finding all the prime numbers up to a specified integer. It was discovered by Indian student S. P. Sundaram in 1934.
Algorithm
St ...
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TurĂĄn sieve
In number theory, the TurĂĄn sieve is a technique for estimating the size of "sifted sets" of positive integers which satisfy a set of conditions which are expressed by congruences. It was developed by PĂĄl TurĂĄn in 1934.
Description
In terms o ...
Cullen prime Cullen may refer to:
Places Canada
*Cullen, Saskatchewan, a former hamlet in Benson No. 35 Rural Municipality
Ireland
*Cullen, County Cork, a village near Boherbue, County Cork
*Cullen, County Tipperary, a small village in County Tipperary
Scotl ...
Mersenne prime
In mathematics, a Mersenne prime is a prime number that is one less than a power of two. That is, it is a prime number of the form for some integer . They are named after Marin Mersenne, a French Minim friar, who studied them in the early 17t ...
Great Internet Mersenne Prime Search
The Great Internet Mersenne Prime Search (GIMPS) is a collaborative project of volunteers who use freely available software to search for Mersenne prime numbers.
GIMPS was founded in 1996 by George Woltman, who also wrote the Prime95 client and ...
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NewmanâShanksâWilliams prime
In mathematics, a NewmanâShanksâWilliams prime (NSW prime) is a prime number ''p'' which can be written in the form
:S_=\frac.
NSW primes were first described by Morris Newman, Daniel Shanks and Hugh C. Williams in 1981 during the study of ...
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Primorial prime In mathematics, a primorial prime is a prime number of the form ''pn''# ± 1, where ''pn''# is the primorial of ''pn'' (i.e. the product of the first ''n'' primes).
Primality tests show that
: ''pn''# â 1 is prime for ''n ...
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Wagstaff prime
In number theory, a Wagstaff prime is a prime number of the form
:
where ''p'' is an odd prime. Wagstaff primes are named after the mathematician Samuel S. Wagstaff Jr.; the prime pages credit François Morain for naming them in a lecture at the ...
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WallâSunâSun prime
In number theory, a WallâSunâSun prime or FibonacciâWieferich prime is a certain kind of prime number which is conjectured to exist, although none are known.
Definition
Let p be a prime number. When each term in the sequence of Fibonac ...
Wilson prime
In number theory, a Wilson prime is a prime number p such that p^2 divides (p-1)!+1, where "!" denotes the factorial function; compare this with Wilson's theorem, which states that every prime p divides (p-1)!+1. Both are named for 18th-century E ...
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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 3. Wolstenholme primes ...
Combinatorial 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 Friedrich Gauss (1777â1855) said, "Math ...
Small set (combinatorics)
In combinatorial mathematics, a large set of positive integers
:S = \
is one such that the infinite sum of the reciprocals
:\frac+\frac+\frac+\frac+\cdots
diverges. A small set is any subset of the positive integers that is not large; that i ...
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ErdĆsâGinzburgâZiv theorem In number theory, zero-sum problems are certain kinds of combinatorial problems about the structure of a finite abelian group. Concretely, given a finite abelian group ''G'' and a positive integer ''n'', one asks for the smallest value of ''k'' suc ...
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Polynomial method
In mathematics, a polynomial is an expression consisting of indeterminates (also called variables) and coefficients, that involves only the operations of addition, subtraction, multiplication, and positive-integer powers of variables. An ex ...
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Van der Waerden's theorem
Van der Waerden's theorem is a theorem in the branch of mathematics called Ramsey theory. Van der Waerden's theorem states that for any given positive integers ''r'' and ''k'', there is some number ''N'' such that if the integers are colored, eac ...
Collatz conjecture
The Collatz conjecture is one of the most famous unsolved problems in mathematics. The conjecture asks whether repeating two simple arithmetic operations will eventually transform every positive integer into 1. It concerns sequences of integ ...
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Gilbreath's conjecture
Gilbreath's conjecture is 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 ter ...
ZnĂĄm's problem
In number theory, ZnĂĄm's problem asks which sets of 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. ZnĂĄm's problem is named after the Slovak mathematician Ć te ...
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 and arithmetic geometry, including algorithms ...
Note: Computational number theory is also known as algorithmic number theory.
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Residue number system
A residue numeral system (RNS) is a numeral system representing integers by their values modulo several pairwise coprime integers called the moduli. This representation is allowed by the Chinese remainder theorem, which asserts that, if is the pro ...
Primality test
A primality test is an algorithm for determining whether an input number is prime. Among other fields of mathematics, it is used for cryptography. Unlike integer factorization, primality tests do not generally give prime factors, only stating whet ...
Sieve of Eratosthenes
In mathematics, the sieve of Eratosthenes is an ancient algorithm for finding all prime numbers up to any given limit.
It does so by iteratively marking as composite (i.e., not prime) the multiples of each prime, starting with the first prime n ...
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Probabilistic algorithm
A randomized algorithm is an algorithm that employs a degree of randomness as part of its logic or procedure. The algorithm typically uses uniformly random bits as an auxiliary input to guide its behavior, in the hope of achieving good performan ...
Euler pseudoprime In arithmetic, an odd composite integer ''n'' is called an Euler pseudoprime to base ''a'', if ''a'' and ''n'' are coprime, and
: a^ \equiv \pm 1\pmod
(where ''mod'' refers to the modulo operation).
The motivation for this definition is the f ...
BaillieâPSW primality test The BaillieâPSW primality test is a probabilistic primality testing algorithm that determines whether a number is composite or is a probable prime. It is named after Robert Baillie, Carl Pomerance, John Selfridge, and Samuel Wagstaff.
The Bailli ...
LucasâLehmer primality test
In mathematics, the LucasâLehmer test (LLT) is a primality test for Mersenne numbers. The test was originally developed by Ădouard Lucas in 1876 and subsequently improved by Derrick Henry Lehmer in the 1930s.
The test
The LucasâLehmer test ...
Integer factorization
In number theory, integer factorization is the decomposition of a composite number into a product of smaller integers. If these factors are further restricted to prime numbers, the process is called prime factorization.
When the numbers are suf ...
Lenstra elliptic curve factorization
The Lenstra elliptic-curve factorization or the elliptic-curve factorization method (ECM) is a fast, sub-exponential running time, algorithm for integer factorization, which employs elliptic curves. For general-purpose factoring, ECM is the th ...
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Quadratic sieve The quadratic sieve algorithm (QS) is an integer factorization algorithm and, in practice, the second fastest method known (after the general number field sieve). It is still the fastest for integers under 100 decimal digits or so, and is considerab ...
General number field sieve
In number theory, the general number field sieve (GNFS) is the most efficient classical algorithm known for factoring integers larger than . Heuristically, its complexity for factoring an integer (consisting of bits) is of the form
:\exp\left( ...
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Shor's algorithm
Shor's algorithm is a quantum algorithm, quantum computer algorithm for finding the prime factors of an integer. It was developed in 1994 by the American mathematician Peter Shor.
On a quantum computer, to factor an integer N , Shor's algorithm ...
Middle-square method
In mathematics and computer science, the middle-square method is a method of generating pseudorandom numbers. In practice it is a highly flawed method for many practical purposes, since its period is usually very short and it has some severe wea ...
ACORN
The acorn, or oaknut, is the nut of the oaks and their close relatives (genera ''Quercus'' and '' Lithocarpus'', in the family Fagaceae). It usually contains one seed (occasionally
two seeds), enclosed in a tough, leathery shell, and borne ...
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ISAAC
Isaac; grc, ጞÏαΏÎș, IsaĂĄk; ar, Ű„ŰłŰÙ°Ù/Ű„ŰłŰۧÙ, Isáž„Äq; am, áá”áá is one of the three patriarchs of the Israelites and an important figure in the Abrahamic religions, including Judaism, Christianity, and Islam. He was the ...
Stream cipher
stream cipher is a symmetric key cipher where plaintext digits are combined with a pseudorandom cipher digit stream (keystream). In a stream cipher, each plaintext digit is encrypted one at a time with the corresponding digit of the keystream ...
see also
List of random number generators
Random number generators are important in many kinds of technical applications, including physics, engineering or mathematical computer studies (e.g., Monte Carlo simulations), cryptography and gambling (on game servers).
This list includes many c ...
Aliquot sequence
In mathematics, an aliquot sequence is a sequence of positive integers in which each term is the sum of the proper divisors of the previous term. If the sequence reaches the number 1, it ends, since the sum of the proper divisors of 1 is 0.
Defi ...
Almost perfect number
In mathematics, an almost perfect number (sometimes also called slightly defective or least deficient number) is a natural number ''n'' such that the sum of all divisors of ''n'' (the sum-of-divisors function ''Ï''(''n'')) is equal to 2''n'' â ...
Quasiperfect number In mathematics, a quasiperfect number is a natural number ''n'' for which the sum of all its divisors (the divisor function ''Ï''(''n'')) is equal to 2''n'' + 1. Equivalently, ''n'' is the sum of its non-trivial divisors (that is, its divisors excl ...
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Perfect number
In number theory, a perfect number is a positive integer that is equal to the sum of its positive divisors, excluding the number itself. For instance, 6 has divisors 1, 2 and 3 (excluding itself), and 1 + 2 + 3 = 6, so 6 is a perfect number.
T ...
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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 sequences, or sociable chains, were discovered and ...
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Collatz conjecture
The Collatz conjecture is one of the most famous unsolved problems in mathematics. The conjecture asks whether repeating two simple arithmetic operations will eventually transform every positive integer into 1. It concerns sequences of integ ...
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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 9045 would be 9 + 0 + 4 + 5 = 18.
Definition
Let n be a natural number. We define the digit ...
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 from the previous iteration to compute a digit su ...
*Digit product dynamics
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Multiplicative digital root
In number theory, the multiplicative digital root of a natural number n in a given number base b is found by multiplying the digits of n together, then repeating this operation until only a single-digit remains, which is called the multiplicati ...
Lychrel number
A Lychrel number is a natural number that cannot form a palindrome through the iterative process of repeatedly reversing its digits and adding the resulting numbers. This process is sometimes called the ''196-algorithm'', after the most famous numb ...
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Perfect digital invariant In number theory, a perfect digital invariant (PDI) is a number in a given number base (b) that is the sum of its own digits each raised to a given power (p). 0 F_ : \mathbb \rightarrow \mathbb is defined as:
:F_(n) = \sum_^ d_i^p.
where k = \lfloo ...
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 arithmetic function, integer-valued functions. German mathematician Carl Friedrich Gauss (1777â ...
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 arithmetic function, integer-valued functions. German mathematician Carl Friedrich Gauss (1777â ...