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Totative
In number theory, a totative of a given positive integer is an integer such that and is coprime to . Euler's totient function φ(''n'') counts the number of totatives of ''n''. The totatives under multiplication modulo ''n'' form the multiplicative group of integers modulo ''n''. Distribution The distribution of totatives has been a subject of study. Paul Erdős conjectured that, writing the totatives of ''n'' as : 0 < a_1 < a_2 \cdots < a_ < n , the mean square gap satisfies : for some constant ''C'', and this was proven by and Hugh Montgomery. See also * |
Number Theory
Number theory is a branch of pure mathematics devoted primarily to the study of the integers and arithmetic functions. Number theorists study prime numbers as well as the properties of mathematical objects constructed from integers (for example, rational numbers), or defined as generalizations of the integers (for example, algebraic integers). Integers can be considered either in themselves or as solutions to equations (Diophantine geometry). Questions in number theory can often be understood through the study of Complex analysis, analytical objects, such as the Riemann zeta function, that encode properties of the integers, primes or other number-theoretic objects in some fashion (analytic number theory). One may also study real numbers in relation to rational numbers, as for instance how irrational numbers can be approximated by fractions (Diophantine approximation). Number theory is one of the oldest branches of mathematics alongside geometry. One quirk of number theory is ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Coprime
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. Consequently, any prime number that divides does not divide , and vice versa. This is equivalent to their greatest common divisor (GCD) being 1. One says also ''is prime to'' or ''is coprime with'' . The numbers 8 and 9 are coprime, despite the fact that neither—considered individually—is a prime number, since 1 is their only common divisor. On the other hand, 6 and 9 are not coprime, because they are both divisible by 3. The numerator and denominator of a reduced fraction are coprime, by definition. Notation and testing When the integers and are coprime, the standard way of expressing this fact in mathematical notation is to indicate that their greatest common divisor is one, by the formula or . In their 1989 textbook '' Concrete Mathematics'', Ronald Graham, Donald Knuth, and Oren Patashnik proposed an alte ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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 other words, it is the number of integers in the range for which the greatest common divisor is equal to 1. The integers of this form are sometimes referred to as totatives of . For example, the totatives of are the six numbers 1, 2, 4, 5, 7 and 8. They are all relatively prime to 9, but the other three numbers in this range, 3, 6, and 9 are not, since and . Therefore, . As another example, since for the only integer in the range from 1 to is 1 itself, and . Euler's totient function is a multiplicative function, meaning that if two numbers and are relatively prime, then . This function gives the order of the multiplicative group of integers modulo (the group of units of the ring \Z/n\Z). It is also used for defining the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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''. Hence another name is the group of primitive residue classes modulo ''n''. In the theory of rings, a branch of abstract algebra, it is described as the group of units of the ring of integers modulo ''n''. Here ''units'' refers to elements with a multiplicative inverse, which, in this ring, are exactly those coprime to ''n''. This group, usually denoted (\mathbb/n\mathbb)^\times, is fundamental in number theory. It is used in cryptography, integer factorization, and primality testing. It is an abelian, finite group whose order is given by Euler's totient function: , (\mathbb/n\mathbb)^\times, =\varph ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Paul Erdős
Paul Erdős ( ; 26March 191320September 1996) was a Hungarian mathematician. He was one of the most prolific mathematicians and producers of mathematical conjectures of the 20th century. pursued and proposed problems in discrete mathematics, graph theory, number theory, mathematical analysis, approximation theory, set theory, and probability theory. Much of his work centered on discrete mathematics, cracking many previously unsolved problems in the field. He championed and contributed to Ramsey theory, which studies the conditions in which order necessarily appears. Overall, his work leaned towards solving previously open problems, rather than developing or exploring new areas of mathematics. Erdős published around 1,500 mathematical papers during his lifetime, a figure that remains unsurpassed. He was known both for his social practice of mathematics, working with more than 500 collaborators, and for his eccentric lifestyle; ''Time'' magazine called him "The Oddball's Oddba ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Bob Vaughan
Robert Charles "Bob" Vaughan FRS (born 24 March 1945) is a British mathematician, working in the field of analytic number theory. Life Vaughan was born 24 March 1945. He read mathematics at University College London, earning a bachelor's degree with second class honours in 1966. He completed his PhD in 1970 at the University of London under supervision of Theodor Estermann. He supervised Trevor Wooley's PhD. After postdoctoral research at the University of Nottingham and University of Sheffield, he became a lecturer in 1972 at Imperial College London. He was promoted to reader in 1976 and professor in 1980, and headed the Pure Mathematics Section from 1988 to 1990. Since 1999, he has been Professor at Pennsylvania State University. Awards Vaughan was a 1979 recipient of the Junior Berwick Prize. Since 1990 Vaughan has been a Fellow of the Royal Society. In 2012, he became a fellow of the American Mathematical Society The American Mathematical Society (AMS) is an asso ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hugh Montgomery (mathematician)
Hugh Lowell Montgomery (born 1944) is an American mathematician, working in the fields of analytic number theory and mathematical analysis. He is the namesake of Montgomery's pair correlation conjecture on the zeros of the Riemann zeta function, is known for his development of large sieve methods, and is the author of multiple books on number theory and analysis. He is a professor emeritus at the University of Michigan. Education and career Montgomery was born on August 26, 1944 in Muncie, Indiana. He was an undergraduate at the University of Illinois Urbana-Champaign. On graduating in 1966, he became a Marshall scholar at the University of Cambridge in England. There, he became a Fellow of Trinity College, Cambridge in 1969, and completed his Ph.D. in 1972. His dissertation, ''Topics in Multiplicative Number Theory'', was supervised by Harold Davenport. He became an assistant professor of mathematics at the University of Michigan in 1972. He was quickly promoted, to associ ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Reduced Residue System
In mathematics, a subset ''R'' of the integers is called a reduced residue system modulo ''n'' if: #gcd(''r'', ''n'') = 1 for each ''r'' in ''R'', #''R'' contains φ(''n'') elements, #no two elements of ''R'' are congruent modulo ''n''. Here φ denotes Euler's totient function. A reduced residue system modulo ''n'' can be formed from a complete residue system modulo ''n'' by removing all integers not relatively prime to ''n''. For example, a complete residue system modulo 12 is . The so-called totatives 1, 5, 7 and 11 are the only integers in this set which are relatively prime to 12, and so the corresponding reduced residue system modulo 12 is . The cardinality of this set can be calculated with the totient function: φ(12) = 4. Some other reduced residue systems modulo 12 are: * * * * Facts *Every number in a reduced residue system modulo ''n'' is a generator for the additive group of integers modulo ''n''. *A reduced residue system modulo ''n'' is a group under multipli ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Springer-Verlag
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. Originally founded in 1842 in Berlin, it expanded internationally in the 1960s, and through mergers in the 1990s and a sale to venture capitalists it fused with Wolters Kluwer and eventually became part of Springer Nature in 2015. Springer has major offices in Berlin, Heidelberg, Dordrecht, and New York City. History Julius Springer founded Springer-Verlag in Berlin in 1842 and his son Ferdinand Springer grew it from a small firm of 4 employees into Germany's then second-largest academic publisher with 65 staff in 1872.Chronology ". Springer Science+Business Media. In 1964, Springer expanded its business internationally, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
