Table of Contents
80 relations: An Introduction to the Theory of Numbers, Andrzej Schinzel, Annals of Mathematics, Arnold Walfisz, Big O notation, Bijection, Carl Friedrich Gauss, Carmichael function, Carmichael's totient function conjecture, Chinese remainder theorem, Concrete Mathematics, Coprime integers, Cyclic group, D. C. Heath and Company, D. H. Lehmer, Dedekind psi function, Dense set, Deutscher Verlag der Wissenschaften, Dirichlet series, Dirichlet's theorem on arithmetic progressions, Discrete Fourier transform, Disquisitiones Arithmeticae, Divisor function, Dover Publications, Duffin–Schaeffer theorem, Euler's constant, Euler's totient function, Ferdinand Rudio, Fermat number, Fermat's little theorem, Fundamental theorem of arithmetic, Greatest common divisor, Highly composite number, Inclusion–exclusion principle, Integer factorization, Inverse function, Ivan Vinogradov, James Joseph Sylvester, Jean-Louis Nicolas, Jordan's totient function, Journal of Number Theory, Lagrange's theorem (group theory), Lambert series, Least common multiple, Leonhard Euler, Limit inferior and limit superior, Möbius function, Möbius inversion formula, MIT Press, Multiplicative function, ... Expand index (30 more) »
- Leonhard Euler
- Multiplicative functions
An Introduction to the Theory of Numbers
An Introduction to the Theory of Numbers is a classic textbook in the field of number theory, by G. H. Hardy and E. M. Wright. Euler's totient function and an Introduction to the Theory of Numbers are number theory.
See Euler's totient function and An Introduction to the Theory of Numbers
Andrzej Schinzel
Andrzej Bobola Maria Schinzel (5 April 1937 – 21 August 2021) was a Polish mathematician studying mainly number theory.
See Euler's totient function and Andrzej Schinzel
Annals of Mathematics
The Annals of Mathematics is a mathematical journal published every two months by Princeton University and the Institute for Advanced Study.
See Euler's totient function and Annals of Mathematics
Arnold Walfisz
Arnold Walfisz (2 July 1892 – 29 May 1962) was a Jewish-Polish mathematician working in analytic number theory.
See Euler's totient function and Arnold Walfisz
Big O notation
Big O notation is a mathematical notation that describes the limiting behavior of a function when the argument tends towards a particular value or infinity.
See Euler's totient function and Big O notation
Bijection
A bijection, bijective function, or one-to-one correspondence between two mathematical sets is a function such that each element of the first set (the domain) is mapped to exactly one element of the second set (the codomain).
See Euler's totient function and Bijection
Carl Friedrich Gauss
Johann Carl Friedrich Gauss (Gauß; Carolus Fridericus Gauss; 30 April 177723 February 1855) was a German mathematician, astronomer, geodesist, and physicist who contributed to many fields in mathematics and science.
See Euler's totient function and Carl Friedrich Gauss
Carmichael function
In number theory, a branch of mathematics, the Carmichael function of a positive integer is the smallest member of the set of positive integers having the property that holds for every integer coprime to. Euler's totient function and Carmichael function are modular arithmetic.
See Euler's totient function and Carmichael function
Carmichael's totient function conjecture
In mathematics, Carmichael's totient function conjecture concerns the multiplicity of values of Euler's totient function φ(n), which counts the number of integers less than and coprime to n. It states that, for every n there is at least one other integer m ≠ n such that φ(m). Euler's totient function and Carmichael's totient function conjecture are multiplicative functions.
See Euler's totient function and Carmichael's totient function conjecture
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 these integers, under the condition that the divisors are pairwise coprime (no two divisors share a common factor other than 1). Euler's totient function and Chinese remainder theorem are modular arithmetic.
See Euler's totient function and Chinese remainder theorem
Concrete Mathematics
Concrete Mathematics: A Foundation for Computer Science, by Ronald Graham, Donald Knuth, and Oren Patashnik, first published in 1989, is a textbook that is widely used in computer-science departments as a substantive but light-hearted treatment of the analysis of algorithms.
See Euler's totient function and Concrete Mathematics
Coprime integers
In number theory, two integers and are coprime, relatively prime or mutually prime if the only positive integer that is a divisor of both of them is 1. Euler's totient function and coprime integers are number theory.
See Euler's totient function and Coprime integers
Cyclic group
In abstract algebra, a cyclic group or monogenous group is a group, denoted Cn (also frequently \Zn or Zn, not to be confused with the commutative ring of p-adic numbers), that is generated by a single element.
See Euler's totient function and Cyclic group
D. C. Heath and Company
D.
See Euler's totient function and D. C. Heath and Company
D. H. Lehmer
Derrick Henry "Dick" Lehmer (February 23, 1905 – May 22, 1991), almost always cited as D.H. Lehmer, was an American mathematician significant to the development of computational number theory.
See Euler's totient function and D. H. Lehmer
Dedekind psi function
In number theory, the Dedekind psi function is the multiplicative function on the positive integers defined by where the product is taken over all primes p dividing n. (By convention, \psi(1), which is the empty product, has value 1.) The function was introduced by Richard Dedekind in connection with modular functions. Euler's totient function and Dedekind psi function are multiplicative functions.
See Euler's totient function and Dedekind psi function
Dense set
In topology and related areas of mathematics, a subset A of a topological space X is said to be dense in X if every point of X either belongs to A or else is arbitrarily "close" to a member of A — for instance, the rational numbers are a dense subset of the real numbers because every real number either is a rational number or has a rational number arbitrarily close to it (see Diophantine approximation).
See Euler's totient function and Dense set
Deutscher Verlag der Wissenschaften
Deutscher Verlag der Wissenschaften (DVW) (English: German Publisher of Sciences) was a scientific publishing house in the former German Democratic Republic (GDR/DDR).
See Euler's totient function and Deutscher Verlag der Wissenschaften
Dirichlet series
In mathematics, a Dirichlet series is any series of the form \sum_^\infty \frac, where s is complex, and a_n is a complex sequence.
See Euler's totient function and Dirichlet series
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 also a positive integer.
See Euler's totient function and Dirichlet's theorem on arithmetic progressions
Discrete Fourier transform
In mathematics, the discrete Fourier transform (DFT) converts a finite sequence of equally-spaced samples of a function into a same-length sequence of equally-spaced samples of the discrete-time Fourier transform (DTFT), which is a complex-valued function of frequency.
See Euler's totient function and Discrete Fourier transform
Disquisitiones Arithmeticae
Disquisitiones Arithmeticae (Latin for Arithmetical Investigations) is a textbook on number theory written in Latin by Carl Friedrich Gauss in 1798, when Gauss was 21, and published in 1801, when he was 24. Euler's totient function and Disquisitiones Arithmeticae are number theory.
See Euler's totient function and Disquisitiones Arithmeticae
Divisor function
In mathematics, and specifically in number theory, a divisor function is an arithmetic function related to the divisors of an integer. Euler's totient function and divisor function are multiplicative functions and number theory.
See Euler's totient function and Divisor function
Dover Publications
Dover Publications, also known as Dover Books, is an American book publisher founded in 1941 by Hayward and Blanche Cirker.
See Euler's totient function and Dover Publications
Duffin–Schaeffer theorem
The Koukoulopoulos–Maynard theorem, also known as the Duffin-Schaeffer conjecture, is a theorem in mathematics, specifically, the Diophantine approximation proposed as a conjecture by R. J. Duffin and A. C. Schaeffer in 1941 and proven in 2019 by Dimitris Koukoulopoulos and James Maynard.
See Euler's totient function and Duffin–Schaeffer theorem
Euler's constant
Euler's constant (sometimes called the Euler–Mascheroni constant) is a mathematical constant, usually denoted by the lowercase Greek letter gamma, defined as the limiting difference between the harmonic series and the natural logarithm, denoted here by: \begin \gamma &. Euler's totient function and Euler's constant are Leonhard Euler.
See Euler's totient function and Euler's constant
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. Euler's totient function and Euler's totient function are algebra, Leonhard Euler, modular arithmetic, multiplicative functions and number theory.
See Euler's totient function and Euler's totient function
Ferdinand Rudio
Ferdinand Rudio (born 2 August 1856 in Wiesbaden, died 21 June 1929 in Zurich) was a German and Swiss mathematician and historian of mathematics.
See Euler's totient function and Ferdinand Rudio
Fermat number
In mathematics, a Fermat number, named after Pierre de Fermat, the first known to have studied them, is a positive integer of the form:F_.
See Euler's totient function and Fermat number
Fermat's little theorem
In number theory, Fermat's little theorem states that if is a prime number, then for any integer, the number is an integer multiple of. Euler's totient function and Fermat's little theorem are modular arithmetic.
See Euler's totient function and Fermat's little theorem
Fundamental theorem of arithmetic
In mathematics, the fundamental theorem of arithmetic, also called the unique factorization theorem and prime factorization theorem, states that every integer greater than 1 can be represented uniquely as a product of prime numbers, up to the order of the factors.
See Euler's totient function and Fundamental theorem of arithmetic
Greatest common divisor
In mathematics, the greatest common divisor (GCD) of two or more integers, which are not all zero, is the largest positive integer that divides each of the integers. Euler's totient function and greatest common divisor are multiplicative functions.
See Euler's totient function and Greatest common divisor
Highly composite number
A highly composite number is a positive integer that has more divisors than any smaller positive integer.
See Euler's totient function and Highly composite number
Inclusion–exclusion principle
In combinatorics, a branch of 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).
See Euler's totient function and Inclusion–exclusion principle
Integer factorization
In number theory, integer factorization is the decomposition of a positive integer into a product of integers.
See Euler's totient function and Integer factorization
Inverse function
In mathematics, the inverse function of a function (also called the inverse of) is a function that undoes the operation of.
See Euler's totient function and Inverse function
Ivan Vinogradov
Ivan Matveevich Vinogradov (a; 14 September 1891 – 20 March 1983) was a Soviet mathematician, who was one of the creators of modern analytic number theory, and also a dominant figure in mathematics in the USSR.
See Euler's totient function and Ivan Vinogradov
James Joseph Sylvester
James Joseph Sylvester (3 September 1814 – 15 March 1897) was an English mathematician.
See Euler's totient function and James Joseph Sylvester
Jean-Louis Nicolas
Jean-Louis Nicolas is a French number theorist.
See Euler's totient function and Jean-Louis Nicolas
Jordan's totient function
In number theory, Jordan's totient function, denoted as J_k(n), where k is a positive integer, is a function of a positive integer, n, that equals the number of k-tuples of positive integers that are less than or equal to n and that together with n form a coprime set of k+1 integers Jordan's totient function is a generalization of Euler's totient function, which is the same as J_1(n). Euler's totient function and Jordan's totient function are modular arithmetic and multiplicative functions.
See Euler's totient function and Jordan's totient function
Journal of Number Theory
The Journal of Number Theory (JNT) is a monthly peer-reviewed scientific journal covering all aspects of number theory. Euler's totient function and journal of Number Theory are number theory.
See Euler's totient function and Journal of Number Theory
Lagrange's theorem (group theory)
In the mathematical field of group theory, Lagrange's theorem is a theorem that states that for any finite group, the order (number of elements) of every subgroup of divides the order of.
See Euler's totient function and Lagrange's theorem (group theory)
Lambert series
In mathematics, a Lambert series, named for Johann Heinrich Lambert, is a series taking the form It can be resummed formally by expanding the denominator: where the coefficients of the new series are given by the Dirichlet convolution of an with the constant function 1(n).
See Euler's totient function and Lambert series
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, is the smallest positive integer that is divisible by both a and b. Since division of integers by zero is undefined, this definition has meaning only if a and b are both different from zero. Euler's totient function and least common multiple are number theory.
See Euler's totient function and Least common multiple
Leonhard Euler
Leonhard Euler (15 April 170718 September 1783) was a Swiss mathematician, physicist, astronomer, geographer, logician, and engineer who founded the studies of graph theory and topology and made pioneering and influential discoveries in many other branches of mathematics such as analytic number theory, complex analysis, and infinitesimal calculus.
See Euler's totient function and Leonhard Euler
Limit inferior and limit superior
In mathematics, the limit inferior and limit superior of a sequence can be thought of as limiting (that is, eventual and extreme) bounds on the sequence.
See Euler's totient function and Limit inferior and limit superior
Möbius function
The Möbius function is a multiplicative function in number theory introduced by the German mathematician August Ferdinand Möbius (also transliterated Moebius) in 1832. Euler's totient function and Möbius function are multiplicative functions.
See Euler's totient function and Möbius function
Möbius inversion formula
In mathematics, the classic Möbius inversion formula is a relation between pairs of arithmetic functions, each defined from the other by sums over divisors.
See Euler's totient function and Möbius inversion formula
MIT Press
The MIT Press is a university press affiliated with the Massachusetts Institute of Technology (MIT) in Cambridge, Massachusetts.
See Euler's totient function and MIT Press
Multiplicative function
In number theory, a multiplicative function is an arithmetic function f(n) of a positive integer n with the property that f(1). Euler's totient function and multiplicative function are multiplicative functions and number theory.
See Euler's totient function and Multiplicative function
Multiplicative group of integers modulo n
In modular arithmetic, the integers coprime (relatively prime) to n from the set \ of n non-negative integers form a group under multiplication modulo ''n'', called the multiplicative group of integers modulo n. Equivalently, the elements of this group can be thought of as the congruence classes, also known as residues modulo n, that are coprime to n. Euler's totient function and multiplicative group of integers modulo n are modular arithmetic.
See Euler's totient function and Multiplicative group of integers modulo n
Multiplicative inverse
In mathematics, a multiplicative inverse or reciprocal for a number x, denoted by 1/x or x−1, is a number which when multiplied by x yields the multiplicative identity, 1.
See Euler's totient function and Multiplicative inverse
Nikolay Korobov
Nikolai Mikhailovich Korobov (Коробов Николай Михайлович; November 23, 1917 – October 25, 2004) was a Soviet mathematician specializing in number theory and numerical analysis.
See Euler's totient function and Nikolay Korobov
Nontotient
In number theory, a nontotient is a positive integer n which is not a totient number: it is not in the range of Euler's totient function φ, that is, the equation φ(x).
See Euler's totient function and Nontotient
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 functions.
See Euler's totient function and Number theory
Order (group theory)
In mathematics, the order of a finite group is the number of its elements.
See Euler's totient function and Order (group theory)
Oxford University Press
Oxford University Press (OUP) is the publishing house of the University of Oxford.
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Phi
Phi (uppercase Φ, lowercase φ or ϕ; ϕεῖ pheî; Modern Greek: φι fi) is the twenty-first letter of the Greek alphabet.
See Euler's totient function and Phi
Pierre Wantzel
Pierre Laurent Wantzel (5 June 1814 in Paris – 21 May 1848 in Paris) was a French mathematician who proved that several ancient geometric problems were impossible to solve using only compass and straightedge.
See Euler's totient function and Pierre Wantzel
Prentice Hall
Prentice Hall was a major American educational publisher.
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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.
See Euler's totient function and Prime number
Prime number theorem
In mathematics, the prime number theorem (PNT) describes the asymptotic distribution of the prime numbers among the positive integers.
See Euler's totient function and Prime number theorem
Primorial
In mathematics, and more particularly in number theory, primorial, denoted by "#", is a function from natural numbers to natural numbers similar to the factorial function, but rather than successively multiplying positive integers, the function only multiplies prime numbers.
See Euler's totient function and Primorial
Radical of an integer
In number theory, the radical of a positive integer n is defined as the product of the distinct prime numbers dividing n. Each prime factor of n occurs exactly once as a factor of this product: \displaystyle\mathrm(n). Euler's totient function and radical of an integer are multiplicative functions.
See Euler's totient function and Radical of an integer
Ramanujan's sum
In number theory, Ramanujan's sum, usually denoted cq(n), is a function of two positive integer variables q and n defined by the formula where (a, q). Euler's totient function and Ramanujan's sum are number theory.
See Euler's totient function and Ramanujan's sum
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.
See Euler's totient function and Riemann hypothesis
Riemann zeta function
The Riemann zeta function or Euler–Riemann zeta function, denoted by the Greek letter (zeta), is a mathematical function of a complex variable defined as \zeta(s).
See Euler's totient function and Riemann zeta function
Ring (mathematics)
In mathematics, rings are algebraic structures that generalize fields: multiplication need not be commutative and multiplicative inverses need not exist.
See Euler's totient function and Ring (mathematics)
Root of unity
In mathematics, a root of unity, occasionally called a de Moivre number, is any complex number that yields 1 when raised to some positive integer power.
See Euler's totient function and Root of unity
RSA (cryptosystem)
RSA (Rivest–Shamir–Adleman) is a public-key cryptosystem, one of the oldest widely used for secure data transmission.
See Euler's totient function and RSA (cryptosystem)
RSA problem
In cryptography, the RSA problem summarizes the task of performing an RSA private-key operation given only the public key.
See Euler's totient function and RSA problem
Schinzel's hypothesis H
In mathematics, Schinzel's hypothesis H is one of the most famous open problems in the topic of number theory.
See Euler's totient function and Schinzel's hypothesis H
Springer Publishing
Springer Publishing Company is an American publishing company of academic journals and books, focusing on the fields of nursing, gerontology, psychology, social work, counseling, public health, and rehabilitation (neuropsychology).
See Euler's totient function and Springer Publishing
Springer Science+Business Media
Springer Science+Business Media, commonly known as Springer, is a German multinational publishing company of books, e-books and peer-reviewed journals in science, humanities, technical and medical (STM) publishing.
See Euler's totient function and Springer Science+Business Media
Subgroup
In group theory, a branch of mathematics, given a group under a binary operation ∗, a subset of is called a subgroup of if also forms a group under the operation ∗.
See Euler's totient function and Subgroup
Totative
In number theory, a totative of a given positive integer is an integer such that. Euler's totient function and totative are modular arithmetic.
See Euler's totient function and Totative
Totient summatory function
In number theory, the totient summatory function \Phi(n) is a summatory function of Euler's totient function defined by: It is the number of coprime integer pairs.
See Euler's totient function and Totient summatory function
Unit (ring theory)
In algebra, a unit or invertible element of a ring is an invertible element for the multiplication of the ring.
See Euler's totient function and Unit (ring theory)
Upper and lower bounds
In mathematics, particularly in order theory, an upper bound or majorant of a subset of some preordered set is an element of that is every element of.
See Euler's totient function and Upper and lower bounds
Wacław Sierpiński
Wacław Franciszek Sierpiński (14 March 1882 – 21 October 1969) was a Polish mathematician.
See Euler's totient function and Wacław Sierpiński
See also
Leonhard Euler
- 2002 Euler
- Contributions of Leonhard Euler to mathematics
- E (mathematical constant)
- Euclid–Euler theorem
- Euler Committee of the Swiss Academy of Sciences
- Euler characteristic
- Euler equations (fluid dynamics)
- Euler function
- Euler method
- Euler number (physics)
- Euler numbers
- Euler's constant
- Euler's continued fraction formula
- Euler's critical load
- Euler's differential equation
- Euler's formula
- Euler's four-square identity
- Euler's identity
- Euler's pump and turbine equation
- Euler's rotation theorem
- Euler's sum of powers conjecture
- Euler's theorem
- Euler's theorem (differential geometry)
- Euler's totient function
- Euler–Bernoulli beam theory
- Euler–Lagrange equation
- Euler–Lotka equation
- Euler–Maclaurin formula
- Euler–Maruyama method
- Euler–Poisson–Darboux equation
- Euler–Rodrigues formula
- Euler–Tricomi equation
- Eulerian path
- Homogeneous function
- Idoneal number
- Institutiones calculi differentialis
- Institutiones calculi integralis
- Introductio in analysin infinitorum
- Leonhard Euler
- Leonhard Euler Gold Medal
- List of things named after Leonhard Euler
- Lucky numbers of Euler
- Opera Omnia Leonhard Euler
- Proof of the Euler product formula for the Riemann zeta function
Multiplicative functions
- Carmichael's totient function conjecture
- Completely multiplicative function
- Dedekind psi function
- Divisor function
- Euler's totient function
- Greatest common divisor
- Jordan's totient function
- Lehmer's totient problem
- Liouville function
- Möbius function
- Multiplicative function
- Radical of an integer
- Ramanujan tau function
- Unit function
References
Also known as Co-totient, Cototient, Euler Phi Function, Euler Totient function, Euler phi, Euler phi-function, Euler totient, Euler's phi, Euler's phi function, Euler's phi-function, Euler's totient, Euler-Totient Function, Eulers phi function, Oiler's totient function, Phi function, Totient, Totient function, Totient number, Totients, Φ function, Φ(n).