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Erdős–Woods Number
In number theory, a positive integer is said to be an Erdős–Woods number if it has the following property: there exists a positive integer such that in the sequence of consecutive integers, each of the elements has a non-trivial common factor with one of the endpoints. In other words, is an Erdős–Woods number if there exists a positive integer such that for each integer between and , at least one of the greatest common divisors or is greater than . Examples 16 is an Erdős–Woods number because the 15 numbers between 2184 and each share a prime factor with one of and These 15 numbers and their shared prime factor(s) are: The first Erdős–Woods numbers are Although all of these initial numbers are even, odd Erdős–Woods numbers also exist. They include Prime partitions The Erdős–Woods numbers can be characterized in terms of certain partitions of the prime numbers. A number is an Erdős–Woods number if and only if the prime numbers less than c ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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] |
86 (number)
86 (eighty-six) is the natural number following 85 (number), 85 and preceding 87 (number), 87. In mathematics 86 is: * nontotient and a noncototient. * the 25th distinct semiprime and the 13th of the form (2.q). * together with 85 (number) , 85 and 87 (number), 87, forms the middle semiprime in the 2nd cluster of three consecutive semiprimes; the first comprising 33 (number), 33, 34 (number), 34, 35 (number), 35. * an Erdős–Woods number, since it is possible to find sequences of 86 consecutive integers such that each inner member shares a factor with either the first or the last member. * a happy number and a self number in base 10. * with an aliquot sum of 46 (number), 46; itself a semiprime, within an aliquot sequence of seven members (86,46 (number), 46,26 (number), 26,16 (number), 16,15 (number), 15,9 (number), 9,4 (number), 4,3 (number), 3,1 (number), 1,0) in the Prime 3 (number), 3-aliquot tree. It appears in the Padovan sequence, preceded by the terms 37, 49, 65 (it is t ... [...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] |
George Szekeres
George Szekeres AM FAA (; 29 May 1911 – 28 August 2005) was a Hungarian–Australian mathematician. Early years Szekeres was born in Budapest, Hungary, as Szekeres György and received his degree in chemistry at the Technical University of Budapest. He worked for six years in Budapest as an analytical chemist. He married Esther Klein in 1937.Obituary The Sydney Morning Herald Being , the family had to escape from the persecution so Szekeres took a job in Shanghai, China. There they lived through World War II, the Japanese occupation an ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Subbayya Sivasankaranarayana Pillai
Subbayya Sivasankaranarayana Pillai (5 April 1901 – 31 August 1950) was an Indian mathematician specialising in number theory. His contribution to Waring's problem was described in 1950 by K. S. Chandrasekharan as "almost certainly his best piece of work and one of the very best achievements in Indian Mathematics since Ramanujan". Biography Subbayya Sivasankaranarayana Pillai was born to parents Subbayya Pillai and Gomati Ammal. His mother died a year after his birth and his father when Pillai was in his last year at school. Pillai did his intermediate course and B.Sc. Mathematics in the Scott Christian College at Nagercoil and managed to earn a B.A. degree from Maharaja's college, Trivandrum. In 1927, Pillai was awarded a research fellowship at the University of Madras to work among professors K. Ananda Rau and Ramaswamy S. Vaidyanathaswamy. He was from 1929 to 1941 at Annamalai University where he worked as a lecturer. It was in Annamalai University that he did his ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
John Selfridge
John Lewis Selfridge (February 17, 1927 – October 31, 2010) was an American mathematician who contributed to the fields of analytic number theory, computational number theory, and combinatorics. Education Selfridge received his Ph.D. in 1958 from the University of California, Los Angeles under the supervision of Theodore Motzkin. Career Selfridge served on the faculties of the University of Illinois at Urbana-Champaign and Northern Illinois University (NIU) from 1971 to 1991 (retirement), chairing the NIU Department of Mathematical Sciences 1972–1976 and 1986–1990. He was executive editor of '' Mathematical Reviews'' from 1978 to 1986, overseeing the computerization of its operations. He was a founder of the Number Theory Foundation, which has named its Selfridge prize in his honour. Research In 1962, he proved that 78,557 is a Sierpinski number; he showed that, when ''k'' = 78,557, all numbers of the form ''k''2''n'' + 1 have a factor in t ... [...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] |
Prime Number
A prime number (or a prime) is a natural number greater than 1 that is not a Product (mathematics), 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 of writing it as a product, or , involve 5 itself. However, 4 is composite because it is a product (2 × 2) in which both numbers are smaller than 4. Primes are central in number theory because of the fundamental theorem of arithmetic: every natural number greater than 1 is either a prime itself or can be factorization, factorized as a product of primes that is unique up to their order. The property of being prime is called primality. A simple but slow primality test, method of checking the primality of a given number , called trial division, tests whether is a multiple of any integer between 2 and . Faster algorithms include the Miller–Rabin primality test, which is fast but has a small chance of error ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
116 (number)
116 (one hundred [and] sixteen) is the natural number following 115 (number), 115 and preceding 117 (number), 117. In mathematics 116 is a noncototient, meaning that there is no solution to the equation , where stands for Euler's totient function. 116! + 1 is a factorial prime. There are 116 ternary Lyndon words of length six, and 116 irreducible polynomials of degree six over a three-element field, which form the basis of a free Lie algebra of dimension 116. There are 116 different ways of partitioning the numbers from 1 through 5 into subsets in such a way that, for every ''k'', the union of the first ''k'' subsets is a consecutive sequence of integers. There are 116 different 6×6 Costas arrays.. See also *116 (other) References {{DEFAULTSORT:116 (Number) Integers ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
112 (number)
112 (one hundred [and] twelve) is the natural number following 111 (number), 111 and preceding 113 (number), 113. Mathematics 112 is an abundant number, a heptagonal number, and a Harshad number. 112 is the number of connected graphs on 6 unlabeled nodes. If an equilateral triangle has sides of length 112, then it contains an interior point at integer distances 57, 65, and 73 from its vertices. This is the smallest possible side length of an equilateral triangle that contains a point at integer distances from the vertices.Wells, D. ''The Penguin Dictionary of Curious and Interesting Numbers'' London: Penguin Group. (1987), page 119 See also * 112 (other) References Integers {{Num-stub ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
106 (number)
106 (one hundred ndsix) is the natural number following 105 and preceding 107 107 may refer to: *107 (number), the number *AD 107, a year in the 2nd century AD *107 BC, a year in the 2nd century BC *107 (New Jersey bus) *107 Camilla, a main-belt asteroid *Peugeot 107, a city car See also *10/7 (other) *Bohrium, .... In mathematics 106 is a centered pentagonal number, a centered heptagonal number, and a regular 19-gonal number. There are 106 mathematical trees with ten vertices. See also * 106 (other) References Integers {{Num-stub ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
96 (number)
96 (ninety-six) is the natural number following 95 (number), 95 and preceding 97 (number), 97. It is a number that Strobogrammatic number, appears the same when rotated by 180 degrees. In mathematics 96 is: * an octagonal number. * a refactorable number. * an untouchable number. * a semiperfect number since it is a multiple of 6. * an abundant number since the sum of its proper divisors is greater than 96. * the fourth Granville number and the second non-perfect Granville number. The next Granville number is 126 (number), 126, the previous being 24 (number), 24. * the sum of Euler's totient function φ(''x'') over the first seventeen integers. * Strobogrammatic number, strobogrammatic in bases 10 (9610), 11 (8811) and 95 (1195). * Palindromic number, palindromic in bases 11 (8811), 15 (6615), 23 (4423), 31 (3331), 47 (2247) and 95 (1195). * an Erdős–Woods number, since it is possible to find sequences of 96 consecutive integers such that each inner member shares a factor with ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
