|
Meissel–Lehmer Algorithm
The Meissel–Lehmer 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, without actually listing them all, dates from Legendre. He observed from the Sieve of Eratosthenes that : \pi (x) - \pi (x^) + 1 = \lfloor x \rfloor - \sum_ \lfloor x/p_i \rfloor + \sum_ \lfloor x/p_ip_j \rfloor - \ldots where \lfloor x \rfloor is the floor function, which denotes the greatest integer less than or equal to x and the p_i run over all primes \leq x^. Since the evaluation of this sum formula is becoming more and more complex and confusing for large x, Meissel tried to simplify the counting of the numbers in the Sieve of Eratosthenes. He and Lehmer introduced therefore certain sieve functions, which are detailed below. Key functions Let p_1, p_2, \ldots, p_n be the first ''n'' primes. For natural number ''a'' ≥ 1, def ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Ernst Meissel
Daniel Friedrich Ernst Meissel (31 July 1826, Eberswalde, Brandenburg Province – 11 March 1895, Kiel) was a German astronomer who contributed to various aspects of number theory. See also *Meissel–Lehmer algorithm *Meissel–Mertens constant External links * 1826 births 1895 deaths 19th-century German astronomers 19th-century German mathematicians Number theorists {{Germany-astronomer-stub ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Derrick Henry 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. Lehmer refined Édouard Lucas' work in the 1930s and devised the Lucas–Lehmer primality test, Lucas–Lehmer test for Mersenne primes. His peripatetic career as a Number theory, number theorist, with him and his wife taking numerous types of work in the United States and abroad to support themselves during the Great Depression, fortuitously brought him into the center of research into early electronic computing. Early life Lehmer was born in Berkeley, California, to Derrick Norman Lehmer, a professor of mathematics at the University of California, Berkeley, and Clara Eunice Mitchell. He studied physics and earned a Bachelor degree from UC Berkeley, and continued with graduate studies at the University of Chicago. He and his father worked together on Lehmer sieves. Marriage During his s ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Algorithm
In mathematics and computer science, an algorithm () is a finite sequence of rigorous instructions, typically used to solve a class of specific Computational problem, problems or to perform a computation. Algorithms are used as specifications for performing calculations and data processing. More advanced algorithms can perform automated deductions (referred to as automated reasoning) and use mathematical and logical tests to divert the code execution through various routes (referred to as automated decision-making). Using human characteristics as descriptors of machines in metaphorical ways was already practiced by Alan Turing with terms such as "memory", "search" and "stimulus". In contrast, a Heuristic (computer science), heuristic is an approach to problem solving that may not be fully specified or may not guarantee correct or optimal results, especially in problem domains where there is no well-defined correct or optimal result. As an effective method, an algorithm ca ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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 the growth rate of the prime-counting function. It was conjectured in the end of the 18th century by Gauss and by Legendre to be approximately : \frac x where log is the natural logarithm, in the sense that :\lim_ \frac=1. This statement is the prime number theorem. An equivalent statement is :\lim_\pi(x) / \operatorname(x)=1 where li is the logarithmic integral function. The prime number theorem was first proved in 1896 by Jacques Hadamard and by Charles de la Vallée Poussin independently, using properties of the Riemann zeta function introduced by Riemann in 1859. Proofs of the prime number theorem not using the zeta function or complex analysis were found around 1948 by Atle Selberg and by Paul Erdős (for the most part inde ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Adrien-Marie Legendre
Adrien-Marie Legendre (; ; 18 September 1752 – 9 January 1833) was a French mathematician who made numerous contributions to mathematics. Well-known and important concepts such as the Legendre polynomials and Legendre transformation are named after him. Life Adrien-Marie Legendre was born in Paris on 18 September 1752 to a wealthy family. He received his education at the Collège Mazarin in Paris, and defended his thesis in physics and mathematics in 1770. He taught at the École Militaire in Paris from 1775 to 1780 and at the École Normale from 1795. At the same time, he was associated with the Bureau des Longitudes. In 1782, the Berlin Academy awarded Legendre a prize for his treatise on projectiles in resistant media. This treatise also brought him to the attention of Lagrange. The ''Académie des sciences'' made Legendre an adjoint member in 1783 and an associate in 1785. In 1789, he was elected a Fellow of the Royal Society. He assisted with the Anglo-French Survey ( ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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 number, 2. The multiples of a given prime are generated as a sequence of numbers starting from that prime, with constant difference between them that is equal to that prime.Horsley, Rev. Samuel, F. R. S., "' or, The Sieve of Eratosthenes. Being an account of his method of finding all the Prime Numbers,''Philosophical Transactions'' (1683–1775), Vol. 62. (1772), pp. 327–347 This is the sieve's key distinction from using trial division to sequentially test each candidate number for divisibility by each prime. Once all the multiples of each discovered prime have been marked as composites, the remaining unmarked numbers are primes. The earliest known reference to the sieve ( grc, κόσκινον Ἐρατοσθένους, ''kóskinon Erat ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Floor And Ceiling Functions
In mathematics and computer science, the floor function is the function that takes as input a real number , and gives as output the greatest integer less than or equal to , denoted or . Similarly, the ceiling function maps to the least integer greater than or equal to , denoted or . For example, for floor: , , and for ceiling: , and . Historically, the floor of has been–and still is–called the integral part or integer part of , often denoted (as well as a variety of other notations). However, the same term, ''integer part'', is also used for truncation towards zero, which differs from the floor function for negative numbers. For an integer, . Although and produce graphs that appear exactly alike, they are not the same when the value of x is an exact integer. For example, when =2.0001; . However, if =2, then , while . Notation The ''integral part'' or ''integer part'' of a number ( in the original) was first defined in 1798 by Adrien-Marie Legendre in his ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
IBM 701
The IBM 701 Electronic Data Processing Machine, known as the Defense Calculator while in development, was IBM’s first commercial scientific computer and its first series production mainframe computer, which was announced to the public on May 21, 1952. It was invented and developed by Jerrier Haddad and Nathaniel Rochester based on the IAS machine at Princeton. The IBM 701 was the first computer in the IBM 700/7000 series, which were IBM’s high-end computers until the arrival of the IBM System/360 in 1964. The business-oriented sibling of the 701 was the IBM 702 and a lower-cost general-purpose sibling was the IBM 650, which gained fame as the first mass-produced computer in the world. History IBM 701 competed with Remington Rand's UNIVAC 1103 in the scientific computation market, which had been developed for the NSA, so it was held secret until permission to market it was obtained in 1951. In early 1954, a committee of the Joint Chiefs of Staff requested that the two ma ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Jeffrey Lagarias
Jeffrey Clark Lagarias (born November 16, 1949 in Pittsburgh, Pennsylvania, United States) is a mathematician and professor at the University of Michigan. Education While in high school in 1966, Lagarias studied astronomy at the Summer Science Program. He completed an S.B. and S.M. in Mathematics at the Massachusetts Institute of Technology in 1972. The title of his thesis was "Evaluation of certain character sums". He was a Putnam Fellow at MIT in 1970. He received his Ph.D. in Mathematics from MIT for his thesis "The 4-part of the class group of a quadratic field", in 1974. His advisor for both his masters and Ph.D was Harold Stark. Career In 1975, he joined AT&T Bell Laboratories and eventually became Distinguished Member of Technical Staff. Since 1995, he has been a Technology Consultant at AT&T Research Laboratories. In 2002, he moved to Michigan to work at the University and settle down with his family. While his recent work has been in theoretical computer science, his ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Victor S
The name Victor or Viktor may refer to: * Victor (name), including a list of people with the given name, mononym, or surname Arts and entertainment Film * ''Victor'' (1951 film), a French drama film * ''Victor'' (1993 film), a French short film * ''Victor'' (2008 film), a 2008 TV film about Canadian swimmer Victor Davis * ''Victor'' (2009 film), a French comedy * ''Victor'', a 2017 film about Victor Torres by Brandon Dickerson * ''Viktor'' (film), a 2014 Franco/Russian film Music * ''Victor'' (album), a 1996 album by Alex Lifeson * "Victor", a song from the 1979 album ''Eat to the Beat'' by Blondie Businesses * Victor Talking Machine Company, early 20th century American recording company, forerunner of RCA Records * Victor Company of Japan, usually known as JVC, a Japanese electronics corporation originally a subsidiary of the Victor Talking Machine Company ** Victor Entertainment, or JVCKenwood Victor Entertainment, a Japanese record label ** Victor Interactive So ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Andrew Odlyzko
Andrew Michael Odlyzko (Andrzej Odłyżko) (born 23 July 1949) is a Polish-American mathematician and a former head of the University of Minnesota's Digital Technology Center and of the Minnesota Supercomputing Institute. He began his career in 1975 at Bell Telephone Laboratories, where he stayed for 26 years before joining the University of Minnesota in 2001. Work in mathematics Odlyzko received his B.S. and M.S. in mathematics from the California Institute of Technology and his Ph.D. from the Massachusetts Institute of Technology in 1975. In the field of mathematics he has published extensively on analytic number theory, computational number theory, cryptography, algorithms and computational complexity, combinatorics, probability, and error-correcting codes. In the early 1970s, he was a co-author (with D. Kahaner and Gian-Carlo Rota) of one of the founding papers of the modern umbral calculus. In 1985 he and Herman te Riele disproved the Mertens conjecture. In mathematic ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
