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# Bernoulli number

In mathematics, the Bernoulli numbers are a sequence of rational numbers which occur frequently in number theory. [1]

129 relations: Abraham de Moivre, Ada Lovelace, Agoh–Giuga conjecture, Al-Karaji, Algebraic number, Algorithm, Alternating permutation, Analytical Engine, Ankeny–Artin–Chowla congruence, Antiderivative, Archimedes, Ars Conjectandi, Aryabhata, Asymptotic analysis, Asymptotic expansion, Bernoulli polynomials, Big O notation, Binomial coefficient, Blaise Pascal, Boustrophedon transform, Carl Gustav Jacob Jacobi, Charles Babbage, Chinese remainder theorem, Closed manifold, Closed-form expression, Coefficient, Computational complexity theory, Computer program, Converse relation, Cumulant, Cyclotomic field, Désiré André, Differentiable manifold, Digamma function, Dimension, Dirichlet character, Dirichlet L-function, Don Zagier, Donald Knuth, Ernst Kummer, Euler number, Euler summation, Euler–Maclaurin formula, Eulerian number, Even and odd functions, Exotic sphere, Falling and rising factorials, Faulhaber's formula, Fermat's Last Theorem, Fibonacci Quarterly, ... Expand index (79 more) »

## Abraham de Moivre

Abraham de Moivre (26 May 166727 November 1754) was a French mathematician known for de Moivre's formula, a formula that links complex numbers and trigonometry, and for his work on the normal distribution and probability theory.

Augusta Ada King-Noel, Countess of Lovelace (née Byron; 10 December 1815 – 27 November 1852) was an English mathematician and writer, chiefly known for her work on Charles Babbage's proposed mechanical general-purpose computer, the Analytical Engine.

## Agoh–Giuga conjecture

In number theory the Agoh–Giuga conjecture on the Bernoulli numbers Bk postulates that p is a prime number if and only if It is named after Takashi Agoh and Giuseppe Giuga.

## Al-Karaji

(c. 953 &ndash; c. 1029) was a 10th-century Persian mathematician and engineer who flourished at Baghdad.

## Algebraic number

An algebraic number is any complex number (including real numbers) that is a root of a non-zero polynomial (that is, a value which causes the polynomial to equal 0) in one variable with rational coefficients (or equivalently – by clearing denominators – with integer coefficients).

## Algorithm

In mathematics and computer science, an algorithm is an unambiguous specification of how to solve a class of problems.

## Alternating permutation

In combinatorial mathematics, an alternating permutation (or zigzag permutation) of the set is an arrangement of those numbers so that each entry is alternately greater or less than the preceding entry.

## Analytical Engine

The Analytical Engine was a proposed mechanical general-purpose computer designed by English mathematician and computer pioneer Charles Babbage.

## Ankeny–Artin–Chowla congruence

In number theory, the Ankeny–Artin–Chowla congruence is a result published in 1953 by N. C. Ankeny, Emil Artin and S. Chowla.

## Antiderivative

In calculus, an antiderivative, primitive function, primitive integral or indefinite integral of a function is a differentiable function whose derivative is equal to the original function.

## Archimedes

Archimedes of Syracuse (Ἀρχιμήδης) was a Greek mathematician, physicist, engineer, inventor, and astronomer.

## Ars Conjectandi

Ars Conjectandi (Latin for "The Art of Conjecturing") is a book on combinatorics and mathematical probability written by Jacob Bernoulli and published in 1713, eight years after his death, by his nephew, Niklaus Bernoulli.

## Aryabhata

Aryabhata (IAST) or Aryabhata I (476–550 CE) was the first of the major mathematician-astronomers from the classical age of Indian mathematics and Indian astronomy.

## Asymptotic analysis

In mathematical analysis, asymptotic analysis, also known as asymptotics, is a method of describing limiting behavior.

## Asymptotic expansion

In mathematics, an asymptotic expansion, asymptotic series or Poincaré expansion (after Henri Poincaré) is a formal series of functions which has the property that truncating the series after a finite number of terms provides an approximation to a given function as the argument of the function tends towards a particular, often infinite, point.

## Bernoulli polynomials

In mathematics, the Bernoulli polynomials, named after Jacob Bernoulli, occur in the study of many special functions and, in particular the Riemann zeta function and the Hurwitz zeta function.

## Big O notation

Big O notation is a mathematical notation that describes the limiting behaviour of a function when the argument tends towards a particular value or infinity.

## Binomial coefficient

In mathematics, any of the positive integers that occurs as a coefficient in the binomial theorem is a binomial coefficient.

## Blaise Pascal

Blaise Pascal (19 June 1623 – 19 August 1662) was a French mathematician, physicist, inventor, writer and Catholic theologian.

## Boustrophedon transform

In mathematics, the boustrophedon transform is a procedure which maps one sequence to another.

## Carl Gustav Jacob Jacobi

Carl Gustav Jacob Jacobi (10 December 1804 – 18 February 1851) was a German mathematician, who made fundamental contributions to elliptic functions, dynamics, differential equations, and number theory.

## Charles Babbage

Charles Babbage (26 December 1791 – 18 October 1871) was an English polymath.

## Chinese remainder theorem

The Chinese remainder theorem is a theorem of number theory, which states that if one knows the remainders of the Euclidean division of an integer by several integers, then one can determine uniquely the remainder of the division of by the product of these integers, under the condition that the divisors are pairwise coprime.

## Closed manifold

In mathematics, a closed manifold is a type of topological space, namely a compact manifold without boundary.

## Closed-form expression

In mathematics, a closed-form expression is a mathematical expression that can be evaluated in a finite number of operations.

## Coefficient

In mathematics, a coefficient is a multiplicative factor in some term of a polynomial, a series or any expression; it is usually a number, but may be any expression.

## Computational complexity theory

Computational complexity theory is a branch of the theory of computation in theoretical computer science that focuses on classifying computational problems according to their inherent difficulty, and relating those classes to each other.

## Computer program

A computer program is a collection of instructions for performing a specific task that is designed to solve a specific class of problems.

## Converse relation

In mathematics, the converse relation, or transpose, of a binary relation is the relation that occurs when the order of the elements is switched in the relation.

## Cumulant

In probability theory and statistics, the cumulants of a probability distribution are a set of quantities that provide an alternative to the moments of the distribution.

## Cyclotomic field

In number theory, a cyclotomic field is a number field obtained by adjoining a complex primitive root of unity to, the field of rational numbers.

## Désiré André

Désiré André (André Antoine Désiré) (March 29, 1840, Lyon – September 12, 1917, Paris) was a French mathematician, best known for his work on Catalan numbers and alternating permutations.

## Differentiable manifold

In mathematics, a differentiable manifold (also differential manifold) is a type of manifold that is locally similar enough to a linear space to allow one to do calculus.

## Digamma function

In mathematics, the digamma function is defined as the logarithmic derivative of the gamma function: It is the first of the polygamma functions.

## Dimension

In physics and mathematics, the dimension of a mathematical space (or object) is informally defined as the minimum number of coordinates needed to specify any point within it.

## 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 characters are used to define Dirichlet ''L''-functions, which are meromorphic functions with a variety of interesting analytic properties.

## Dirichlet L-function

In mathematics, a Dirichlet L-series is a function of the form Here χ is a Dirichlet character and s a complex variable with real part greater than 1.

## Don Zagier

Don Bernard Zagier (born 29 June 1951) is an American mathematician whose main area of work is number theory.

## Donald Knuth

Donald Ervin Knuth (born January 10, 1938) is an American computer scientist, mathematician, and professor emeritus at Stanford University.

## Ernst Kummer

Ernst Eduard Kummer (29 January 1810 – 14 May 1893) was a German mathematician.

## Euler number

In mathematics, the Euler numbers are a sequence En of integers defined by the Taylor series expansion where is the hyperbolic cosine.

## Euler summation

In the mathematics of convergent and divergent series, Euler summation is a summability method.

## Euler–Maclaurin formula

In mathematics, the Euler–Maclaurin formula provides a powerful connection between integrals (see calculus) and sums.

## Eulerian number

In combinatorics, the Eulerian number A(n, m), is the number of permutations of the numbers 1 to n in which exactly m elements are greater than the previous element (permutations with m "ascents").

## Even and odd functions

In mathematics, even functions and odd functions are functions which satisfy particular symmetry relations, with respect to taking additive inverses.

## Exotic sphere

In differential topology, an exotic sphere is a differentiable manifold M that is homeomorphic but not diffeomorphic to the standard Euclidean n-sphere.

## Falling and rising factorials

In mathematics, the falling factorial (sometimes called the descending factorial, falling sequential product, or lower factorial) is defined as The rising factorial (sometimes called the Pochhammer function, Pochhammer polynomial, ascending factorial, (A reprint of the 1950 edition by Chelsea Publishing Co.) rising sequential product, or upper factorial) is defined as The value of each is taken to be 1 (an empty product) when n.

## Faulhaber's formula

In mathematics, Faulhaber's formula, named after Johann Faulhaber, expresses the sum of the p-th powers of the first n positive integers as a (p + 1)th-degree polynomial function of n, the coefficients involving Bernoulli numbers Bj, in the form submitted by Jacob Bernoulli and published in 1713: where p^\underline.

## Fermat's Last Theorem

In number theory, Fermat's Last Theorem (sometimes called Fermat's conjecture, especially in older texts) states that no three positive integers,, and satisfy the equation for any integer value of greater than 2.

## Fibonacci Quarterly

The Fibonacci Quarterly is a scientific journal on mathematical topics related to the Fibonacci numbers, published four times per year.

## Fundamental theorem of calculus

The fundamental theorem of calculus is a theorem that links the concept of differentiating a function with the concept of integrating a function.

## Generating function

In mathematics, a generating function is a way of encoding an infinite sequence of numbers (an) by treating them as the coefficients of a power series.

## Genocchi number

In mathematics, the Genocchi numbers Gn, named after Angelo Genocchi, are a sequence of integers that satisfy the relation \frac.

## Gottfried Wilhelm Leibniz

Gottfried Wilhelm (von) Leibniz (or; Leibnitz; – 14 November 1716) was a German polymath and philosopher who occupies a prominent place in the history of mathematics and the history of philosophy.

## Harmonic number

In mathematics, the -th harmonic number is the sum of the reciprocals of the first natural numbers: Harmonic numbers are related to the harmonic mean in that the -th harmonic number is also times the reciprocal of the harmonic mean of the first positive integers.

## Harmonic progression (mathematics)

In mathematics, a harmonic progression (or harmonic sequence) is a progression formed by taking the reciprocals of an arithmetic progression.

## Herbrand–Ribet theorem

In mathematics, the Herbrand–Ribet theorem is a result on the class group of certain number fields.

## Hurwitz zeta function

In mathematics, the Hurwitz zeta function, named after Adolf Hurwitz, is one of the many zeta functions.

## Hyperbolic function

In mathematics, hyperbolic functions are analogs of the ordinary trigonometric, or circular, functions.

## Ibn al-Haytham

Hasan Ibn al-Haytham (Latinized Alhazen; full name أبو علي، الحسن بن الحسن بن الهيثم) was an Arab mathematician, astronomer, and physicist of the Islamic Golden Age.

## Ideal class group

In number theory, the ideal class group (or class group) of an algebraic number field is the quotient group where is the group of fractional ideals of the ring of integers of, and is its subgroup of principal ideals.

## Imaginary unit

The imaginary unit or unit imaginary number is a solution to the quadratic equation.

## Implementation

Implementation is the realization of an application, or execution of a plan, idea, model, design, specification, standard, algorithm, or policy.

## Inclusion–exclusion principle

In combinatorics (combinatorial mathematics), the inclusion–exclusion principle is a counting technique which generalizes the familiar method of obtaining the number of elements in the union of two finite sets; symbolically expressed as where A and B are two finite sets and |S| indicates the cardinality of a set S (which may be considered as the number of elements of the set, if the set is finite).

## Integral

In mathematics, an integral assigns numbers to functions in a way that can describe displacement, area, volume, and other concepts that arise by combining infinitesimal data.

## Iverson bracket

In mathematics, the Iverson bracket, named after Kenneth E. Iverson, is a notation that generalises the Kronecker delta.

## Jacob Bernoulli

Jacob Bernoulli (also known as James or Jacques; – 16 August 1705) was one of the many prominent mathematicians in the Bernoulli family.

## Johann Faulhaber

Johann Faulhaber (5 May 1580 – 10 September 1635) was a German mathematician.

## Karl Georg Christian von Staudt

Karl Georg Christian von Staudt (24 January 1798 – 1 June 1867) was a German mathematician who used synthetic geometry to provide a foundation for arithmetic.

## Kronecker delta

In mathematics, the Kronecker delta (named after Leopold Kronecker) is a function of two variables, usually just non-negative integers.

## Kummer's congruence

In mathematics, Kummer's congruences are some congruences involving Bernoulli numbers, found by.

## Kummer–Vandiver conjecture

In mathematics, the Kummer–Vandiver conjecture, or Vandiver conjecture, states that a prime p does not divide the class number hK of the maximal real subfield K.

## Laurent series

In mathematics, the Laurent series of a complex function f(z) is a representation of that function as a power series which includes terms of negative degree.

## Leonhard Euler

Leonhard Euler (Swiss Standard German:; German Standard German:; 15 April 170718 September 1783) was a Swiss mathematician, physicist, astronomer, logician and engineer, who made important and influential discoveries in many branches of mathematics, such as infinitesimal calculus and graph theory, while also making pioneering contributions to several branches such as topology and analytic number theory.

## Leopold Kronecker

Leopold Kronecker (7 December 1823 – 29 December 1891) was a German mathematician who worked on number theory, algebra and logic.

## Logical equivalence

In logic, statements p and q are logically equivalent if they have the same logical content.

## Marcel Riesz

Marcel Riesz (Riesz Marcell; 16 November 1886 – 4 September 1969) was a Hungarian-born mathematician, known for work on summation methods, potential theory, and other parts of analysis, as well as number theory, partial differential equations, and Clifford algebras.

## Mathematics

Mathematics (from Greek μάθημα máthēma, "knowledge, study, learning") is the study of such topics as quantity, structure, space, and change.

## Mersenne prime

In mathematics, a Mersenne prime is a prime number that is one less than a power of two.

## National Institute of Standards and Technology

The National Institute of Standards and Technology (NIST) is one of the oldest physical science laboratories in the United States.

## Noam Elkies

Noam David Elkies (born August 25, 1966) is an American mathematician and professor of mathematics at Harvard University.

## Number theory

Number theory, or in older usage arithmetic, is a branch of pure mathematics devoted primarily to the study of the integers.

## Orientability

In mathematics, orientability is a property of surfaces in Euclidean space that measures whether it is possible to make a consistent choice of surface normal vector at every point.

In mathematics, a p-adic zeta function, or more generally a p-adic L-function, is a function analogous to the Riemann zeta function, or more general ''L''-functions, but whose domain and target are p-adic (where p is a prime number).

In mathematics, the -adic number system for any prime number extends the ordinary arithmetic of the rational numbers in a different way from the extension of the rational number system to the real and complex number systems.

## Parallelizable manifold

In mathematics, a differentiable manifold M of dimension n is called parallelizable if there exist smooth vector fields on the manifold, such that at any point p of M the tangent vectors provide a basis of the tangent space at p. Equivalently, the tangent bundle is a trivial bundle, so that the associated principal bundle of linear frames has a section on M. A particular choice of such a basis of vector fields on M is called a parallelization (or an absolute parallelism) of M.

## Pascal's triangle

In mathematics, Pascal's triangle is a triangular array of the binomial coefficients.

## Philipp Ludwig von Seidel

Philipp Ludwig von Seidel (23 October 1821 in Zweibrücken, Germany – 13 August 1896 in Munich, German Empire) was a German mathematician.

## Pierre de Fermat

Pierre de Fermat (Between 31 October and 6 December 1607 – 12 January 1665) was a French lawyer at the Parlement of Toulouse, France, and a mathematician who is given credit for early developments that led to infinitesimal calculus, including his technique of adequality.

## Poly-Bernoulli number

In mathematics, poly-Bernoulli numbers, denoted as B_^, were defined by M. Kaneko as where Li is the polylogarithm.

## Polynomial

In mathematics, a polynomial is an expression consisting of variables (also called indeterminates) and coefficients, that involves only the operations of addition, subtraction, multiplication, and non-negative integer exponents of variables.

## Prime number

A prime number (or a prime) is a natural number greater than 1 that cannot be formed by multiplying two smaller natural numbers.

## Probability distribution

In probability theory and statistics, a probability distribution is a mathematical function that provides the probabilities of occurrence of different possible outcomes in an experiment.

## Project Gutenberg

Project Gutenberg (PG) is a volunteer effort to digitize and archive cultural works, to "encourage the creation and distribution of eBooks".

## Pythagoras

Pythagoras of Samos was an Ionian Greek philosopher and the eponymous founder of the Pythagoreanism movement.

## Q-analog

In mathematics, a q-analog of a theorem, identity or expression is a generalization involving a new parameter q that returns the original theorem, identity or expression in the limit as.

## Rahul Pandharipande

Rahul Pandharipande (born 1969) is a mathematician who is currently a professor of mathematics at the Swiss Federal Institute of Technology Zürich (ETH) working in algebraic geometry.

## Rational number

In mathematics, a rational number is any number that can be expressed as the quotient or fraction of two integers, a numerator and a non-zero denominator.

## Regular prime

In number theory, a regular prime is a special kind of prime number, defined by Ernst Kummer in 1850 to prove certain cases of Fermat's Last Theorem.

## Riemann hypothesis

In mathematics, the Riemann hypothesis is a conjecture that the Riemann zeta function has its zeros only at the negative even integers and complex numbers with real part.

## Riemann zeta function

The Riemann zeta function or Euler–Riemann zeta function,, is a function of a complex variable s that analytically continues the sum of the Dirichlet series which converges when the real part of is greater than 1.

## Riesz function

In mathematics, the Riesz function is an entire function defined by Marcel Riesz in connection with the Riemann hypothesis, by means of the power series If we set F(x).

## Ronald Graham

Ronald Lewis "Ron" Graham (born October 31, 1935) is an American mathematician credited by the American Mathematical Society as being "one of the principal architects of the rapid development worldwide of discrete mathematics in recent years".

## SageMath

SageMath (previously Sage or SAGE, "System for Algebra and Geometry Experimentation") is a computer algebra system with features covering many aspects of mathematics, including algebra, combinatorics, graph theory, numerical analysis, number theory, calculus and statistics.

## Scientific notation

Scientific notation (also referred to as scientific form or standard index form, or standard form in the UK) is a way of expressing numbers that are too big or too small to be conveniently written in decimal form.

## Seki Takakazu

, also known as,Selin, was a Japanese mathematician and author of the Edo period.

## Sequence

In mathematics, a sequence is an enumerated collection of objects in which repetitions are allowed.

## Simon Plouffe

Simon Plouffe (born June 11, 1956, Saint-Jovite, Quebec) is a mathematician who discovered the Bailey–Borwein–Plouffe formula (BBP algorithm) which permits the computation of the nth binary digit of &pi;, in 1995.

## Special values of L-functions

In mathematics, the study of special values of L-functions is a subfield of number theory devoted to generalising formulae such as the Leibniz formula for pi, namely by the recognition that expression on the left-hand side is also L(1) where L(s) is the Dirichlet L-function for the Gaussian field.

## Square pyramidal number

In mathematics, a pyramid number, or square pyramidal number, is a figurate number that represents the number of stacked spheres in a pyramid with a square base.

## Square-free element

In mathematics, a square-free element is an element r of a unique factorization domain R that is not divisible by a non-trivial square.

## Srinivasa Ramanujan

Srinivasa Ramanujan (22 December 188726 April 1920) was an Indian mathematician who lived during the British Rule in India. Though he had almost no formal training in pure mathematics, he made substantial contributions to mathematical analysis, number theory, infinite series, and continued fractions, including solutions to mathematical problems considered to be unsolvable.

## Stirling numbers of the first kind

In mathematics, especially in combinatorics, Stirling numbers of the first kind arise in the study of permutations.

## Stirling numbers of the second kind

In mathematics, particularly in combinatorics, a Stirling number of the second kind (or Stirling partition number) is the number of ways to partition a set of n objects into k non-empty subsets and is denoted by S(n,k) or \textstyle \lbrace\rbrace.

## Stirling polynomials

In mathematics, the Stirling polynomials are a family of polynomials that generalize important sequences of numbers appearing in combinatorics and analysis, which are closely related to the Stirling numbers, the Bernoulli numbers, and the generalized Bernoulli polynomials.

## Stirling's approximation

In mathematics, Stirling's approximation (or Stirling's formula) is an approximation for factorials.

## Summation

In mathematics, summation (capital Greek sigma symbol: ∑) is the addition of a sequence of numbers; the result is their sum or total.

## Sums of powers

In mathematics and statistics, sums of powers occur in a number of contexts.

## Taylor series

In mathematics, a Taylor series is a representation of a function as an infinite sum of terms that are calculated from the values of the function's derivatives at a single point.

## Thomas Clausen (mathematician)

Thomas Clausen (January 16, 1801, Snogbæk, Sottrup Municipality, Duchy of Schleswig (now Denmark) &ndash; May 23, 1885, Derpt, Imperial Russia (now Estonia)) was a Danish mathematician and astronomer.

## Thomas Harriot

Thomas Harriot (Oxford, c. 1560 – London, 2 July 1621), also spelled Harriott, Hariot or Heriot, was an English astronomer, mathematician, ethnographer and translator who made advances within the scientific field.

## Triangular number

A triangular number or triangle number counts objects arranged in an equilateral triangle, as in the diagram on the right.

## Trigamma function

In mathematics, the trigamma function, denoted, is the second of the polygamma functions, and is defined by It follows from this definition that where is the digamma function.

## Trigonometric functions

In mathematics, the trigonometric functions (also called circular functions, angle functions or goniometric functions) are functions of an angle.

## Umbral calculus

In mathematics before the 1970s, the term umbral calculus referred to the surprising similarity between seemingly unrelated polynomial equations and certain shadowy techniques used to 'prove' them.

## Uniform distribution (continuous)

In probability theory and statistics, the continuous uniform distribution or rectangular distribution is a family of symmetric probability distributions such that for each member of the family, all intervals of the same length on the distribution's support are equally probable.

## 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.

## Wolfram Mathematica

Wolfram Mathematica (usually termed Mathematica) is a modern technical computing system spanning most areas of technical computing — including neural networks, machine learning, image processing, geometry, data science, visualizations, and others.

## Yuri Matiyasevich

Yuri Vladimirovich Matiyasevich, (Ю́рий Влади́мирович Матиясе́вич; born March 2, 1947, in Leningrad) is a Russian mathematician and computer scientist.

## References

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