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Distributed computing
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Distributed computing is a field of computer science that studies distributed systems. A distributed system is a model in which components located on networked computers communicate and coordinate their actions by passing messages, the components interact with each other in order to achieve a common goal. Three significant characteristics of distributed systems are, concurrency of components, lack of a global clock, examples of distributed systems vary from SOA-based systems to massively multiplayer online games to peer-to-peer applications. A computer program that runs in a system is called a distributed program. There are many alternatives for the message passing mechanism, including pure HTTP, RPC-like connectors, Distributed computing also refers to the use of distributed systems to solve computational problems. In distributed computing, a problem is divided into many tasks, each of which is solved by one or more computers, which communicate with each other by message passing. The terms are used in a much wider sense, even referring to autonomous processes that run on the same physical computer. The entities communicate with each other by message passing, a distributed system may have a common goal, such as solving a large computational problem, the user then perceives the collection of autonomous processors as a unit. Other typical properties of distributed systems include the following, The system has to tolerate failures in individual computers. The structure of the system is not known in advance, the system may consist of different kinds of computers and network links, each computer has only a limited, incomplete view of the system. Each computer may know one part of the input. Distributed systems are groups of networked computers, which have the goal for their work. The terms concurrent computing, parallel computing, and distributed computing have a lot of overlap, the same system may be characterized both as parallel and distributed, the processors in a typical distributed system run concurrently in parallel. Parallel computing may be seen as a tightly coupled form of distributed computing. In distributed computing, each processor has its own private memory, Information is exchanged by passing messages between the processors. The figure on the right illustrates the difference between distributed and parallel systems, figure shows a parallel system in which each processor has a direct access to a shared memory. The situation is complicated by the traditional uses of the terms parallel and distributed algorithm that do not quite match the above definitions of parallel. The use of concurrent processes that communicate by message-passing has its roots in operating system architectures studied in the 1960s, the first widespread distributed systems were local-area networks such as Ethernet, which was invented in the 1970s
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PrimeGrid
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PrimeGrid is a distributed computing project for searching for prime numbers of world-record size. It makes use of the Berkeley Open Infrastructure for Network Computing platform, PrimeGrid started in June 2005 under the name Message@home and tried to decipher text fragments hashed with MD5. Message@home was a test to port the BOINC scheduler to Perl to obtain greater portability, after a while the project attempted the RSA factoring challenge trying to factor RSA-640. After RSA-640 was factored by a team in November 2005. With the chance to succeed too small, it discarded the RSA challenges, was renamed to PrimeGrid, at 210,000,000,000 the primegen subproject was stopped. In June 2006, dialog started with Riesel Sieve to bring their project to the BOINC community, PrimeGrid provided PerlBOINC support and Riesel Sieve was successful in implementing their sieve as well as a prime finding application. With collaboration from Riesel Sieve, PrimeGrid was able to implement the LLR application in partnership with another prime finding project, in November 2006, the TPS LLR application was officially released at PrimeGrid. Less than two months later, January 2007, the twin was found by the original manual project. PrimeGrid and TPS then advanced their search for even larger twin primes, the summer of 2007 was very active as the Cullen and Woodall prime searches were launched. In the Fall, more prime searches were added through partnerships with the Prime Sierpinski Problem, additionally, two sieves were added, the Prime Sierpinski Problem combined sieve which includes supporting the Seventeen or Bust sieve, and the combined Cullen/Woodall sieve. In the Fall of 2007, PrimeGrid migrated its systems from PerlBOINC to standard BOINC software, since September 2008, PrimeGrid is also running a Proth prime sieving subproject. In January 2010 the subproject Seventeen or Bust was added, the calculations for the Riesel problem followed in March 2010. In addition, PrimeGrid is helping test for a record Sophie Germain prime. As of March 2016, PrimeGrid is working on or has worked on the projects,321 Prime Search is a continuation of Paul Underwoods 321 Search which looked for primes of the form 3 · 2n −1. PrimeGrid added the +1 form and continues the search up to n = 25M, the search was successful in April 2010 with the finding of the first known AP26,43142746595714191 +23681770 · 23# · n is prime for n =0. 23# = 2·3·5·7·11·13·17·19·23 =223092870, or 23 primorial, is the product of all primes up to 23, PrimeGrid is also running a search for Cullen prime numbers, yielding the two largest known Cullen primes. The first one being the 14th largest known prime at the time of discovery, as of 9 March 2014 PrimeGrid has eliminated 14 values of k from the Riesel problem and is continuing the search to eliminate the 50 remaining numbers. Primegrid then worked with the Twin Prime Search to search for a twin prime at approximately 58700 digits
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Composite number
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A composite number is a positive integer that can be formed by multiplying together two smaller positive integers. Equivalently, it is an integer that has at least one divisor other than 1. Every positive integer is composite, prime, or the unit 1, so the numbers are exactly the numbers that are not prime. For example, the integer 14 is a number because it is the product of the two smaller integers 2 ×7. Likewise, the integers 2 and 3 are not composite numbers because each of them can only be divided by one, every composite number can be written as the product of two or more primes. For example, the composite number 299 can be written as 13 ×23, and the composite number 360 can be written as 23 ×32 ×5, furthermore and this fact is called the fundamental theorem of arithmetic. There are several known primality tests that can determine whether a number is prime or composite, one way to classify composite numbers is by counting the number of prime factors. A composite number with two prime factors is a semiprime or 2-almost prime, a composite number with three distinct prime factors is a sphenic number. In some applications, it is necessary to differentiate between composite numbers with an odd number of prime factors and those with an even number of distinct prime factors. For the latter μ =2 x =1, while for the former μ =2 x +1 = −1, however, for prime numbers, the function also returns −1 and μ =1. For a number n with one or more repeated prime factors, if all the prime factors of a number are repeated it is called a powerful number. If none of its factors are repeated, it is called squarefree. For example,72 =23 ×32, all the factors are repeated. 42 =2 ×3 ×7, none of the factors are repeated. Another way to classify composite numbers is by counting the number of divisors, all composite numbers have at least three divisors. In the case of squares of primes, those divisors are, a number n that has more divisors than any x < n is a highly composite number. Composite numbers have also been called rectangular numbers, but that name can refer to the pronic numbers, numbers that are the product of two consecutive integers. Table of prime factors Integer factorization Canonical representation of a positive integer Sieve of Eratosthenes Fraleigh, a First Course In Abstract Algebra, Reading, Addison-Wesley, ISBN 0-201-01984-1 Herstein, I. N
4.
Prime number
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A prime number is a natural number greater than 1 that has no positive divisors other than 1 and itself. A natural number greater than 1 that is not a number is called a composite number. For example,5 is prime because 1 and 5 are its only positive integer factors, the property of being prime is called primality. A simple but slow method of verifying the primality of a number n is known as trial division. It consists of testing whether n is a multiple of any integer between 2 and n, algorithms much more efficient than trial division have been devised to test the primality of large numbers. Particularly fast methods are available for numbers of forms, such as Mersenne numbers. As of January 2016, the largest known prime number has 22,338,618 decimal digits, there are infinitely many primes, as demonstrated by Euclid around 300 BC. There is no simple formula that separates prime numbers from composite numbers. However, the distribution of primes, that is to say, many questions regarding prime numbers remain open, such as Goldbachs conjecture, and the twin prime conjecture. Such questions spurred the development of branches of number theory. Prime numbers give rise to various generalizations in other domains, mainly algebra, such as prime elements. A natural number is called a number if it has exactly two positive divisors,1 and the number itself. Natural numbers greater than 1 that are not prime are called composite, among the numbers 1 to 6, the numbers 2,3, and 5 are the prime numbers, while 1,4, and 6 are not prime. 1 is excluded as a number, for reasons explained below. 2 is a number, since the only natural numbers dividing it are 1 and 2. Next,3 is prime, too,1 and 3 do divide 3 without remainder, however,4 is composite, since 2 is another number dividing 4 without remainder,4 =2 ·2. 5 is again prime, none of the numbers 2,3, next,6 is divisible by 2 or 3, since 6 =2 ·3. The image at the right illustrates that 12 is not prime,12 =3 ·4, no even number greater than 2 is prime because by definition, any such number n has at least three distinct divisors, namely 1,2, and n
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Sequence
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In mathematics, a sequence is an enumerated collection of objects in which repetitions are allowed. Like a set, it contains members, the number of elements is called the length of the sequence. Unlike a set, order matters, and exactly the elements can appear multiple times at different positions in the sequence. Formally, a sequence can be defined as a function whose domain is either the set of the numbers or the set of the first n natural numbers. The position of an element in a sequence is its rank or index and it depends on the context or of a specific convention, if the first element has index 0 or 1. For example, is a sequence of letters with the letter M first, also, the sequence, which contains the number 1 at two different positions, is a valid sequence. Sequences can be finite, as in these examples, or infinite, the empty sequence is included in most notions of sequence, but may be excluded depending on the context. A sequence can be thought of as a list of elements with a particular order, Sequences are useful in a number of mathematical disciplines for studying functions, spaces, and other mathematical structures using the convergence properties of sequences. In particular, sequences are the basis for series, which are important in differential equations, Sequences are also of interest in their own right and can be studied as patterns or puzzles, such as in the study of prime numbers. There are a number of ways to denote a sequence, some of which are useful for specific types of sequences. One way to specify a sequence is to list the elements, for example, the first four odd numbers form the sequence. This notation can be used for sequences as well. For instance, the sequence of positive odd integers can be written. Listing is most useful for sequences with a pattern that can be easily discerned from the first few elements. Other ways to denote a sequence are discussed after the examples, the prime numbers are the natural numbers bigger than 1, that have no divisors but 1 and themselves. Taking these in their natural order gives the sequence, the prime numbers are widely used in mathematics and specifically in number theory. The Fibonacci numbers are the integer sequence whose elements are the sum of the two elements. The first two elements are either 0 and 1 or 1 and 1 so that the sequence is, for a large list of examples of integer sequences, see On-Line Encyclopedia of Integer Sequences
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Conjecture
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In mathematics, a conjecture is a conclusion or proposition based on incomplete information, for which no proof has been found. Conjectures such as the Riemann hypothesis or Fermats Last Theorem have shaped much of history as new areas of mathematics are developed in order to prove them. In number theory, Fermats Last Theorem states that no three positive integers a, b, and c can satisfy the equation an + bn = cn for any value of n greater than two. This theorem was first conjectured by Pierre de Fermat in 1637 in the margin of a copy of Arithmetica where he claimed he had a proof that was too large to fit in the margin. The first successful proof was released in 1994 by Andrew Wiles, the unsolved problem stimulated the development of algebraic number theory in the 19th century and the proof of the modularity theorem in the 20th century. It is among the most notable theorems in the history of mathematics, two regions are called adjacent if they share a common boundary that is not a corner, where corners are the points shared by three or more regions. For example, in the map of the United States of America, Utah and Arizona are adjacent, but Utah and New Mexico, möbius mentioned the problem in his lectures as early as 1840. The conjecture was first proposed on October 23,1852 when Francis Guthrie, while trying to color the map of counties of England, a number of false proofs and false counterexamples have appeared since the first statement of the four color theorem in 1852. The four color theorem was proven in 1976 by Kenneth Appel and it was the first major theorem to be proved using a computer. Appel and Hakens approach started by showing that there is a set of 1,936 maps. Appel and Haken used a computer program to confirm that each of these maps had this property. Additionally, any map that could potentially be a counterexample must have a portion that looks like one of these 1,936 maps, showing this required hundreds of pages of hand analysis. Appel and Haken concluded that no smallest counterexamples exists because any must contain, yet do not contain and this contradiction means there are no counterexamples at all and that the theorem is therefore true. Initially, their proof was not accepted by all mathematicians because the proof was infeasible for a human to check by hand. Since then the proof has gained acceptance, although doubts remain. The Hauptvermutung of geometric topology is the conjecture that any two triangulations of a triangulable space have a refinement, a single triangulation that is a subdivision of both of them. It was originally formulated in 1908, by Steinitz and Tietze and this conjecture is now known to be false. The non-manifold version was disproved by John Milnor in 1961 using Reidemeister torsion, the manifold version is true in dimensions m ≤3
