1.
Great Internet Mersenne Prime Search
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The Great Internet Mersenne Prime Search is a collaborative project of volunteers who use freely available software to search for Mersenne prime numbers. The GIMPS project was founded by George Woltman, who wrote the software Prime95. Scott Kurowski wrote the PrimeNet Internet server that supports the research to demonstrate Entropia-distributed computing software, GIMPS is registered as Mersenne Research, Inc. Kurowski is Executive Vice President and board director of Mersenne Research Inc, GIMPS is said to be one of the first large scale distributed computing projects over the Internet for research purposes. The project has found a total of fifteen Mersenne primes as of January 2016, the largest known prime as of January 2016 is 274,207,281 −1. This prime was discovered on September 17,2015 by Curtis Cooper at the University of Central Missouri and they also have a trial division phase, used to rapidly eliminate Mersenne numbers with small factors which make up a large proportion of candidates. Pollards p -1 algorithm is used to search for larger factors. The project began in early January 1996, with a program ran on i386 computers. The name for the project was coined by Luther Welsh, one of its earlier searchers, within a few months, several dozen people had joined, and over a thousand by the end of the first year. Joel Armengaud, a participant, discovered the primality of M1,398,269 on November 13,1996, as of March 2013, GIMPS has a sustained aggregate throughput of approximately 137.023 TFLOP/s. In November 2012, GIMPS maintained 95 TFLOP/s, theoretically earning the GIMPS virtual computer a place among the TOP500 most powerful computer systems in the world. Also theoretically, in November 2012, the GIMPS held a rank of 330 in the TOP500, the preceding place was then held by an HP Cluster Platform 3000 BL460c G7 of Hewlett-Packard. As of November 2014 TOP500 results, these old GIMPS numbers would no longer make the list, previously, this was approximately 50 TFLOP/s in early 2010,30 TFLOP/s in mid-2008,20 TFLOP/s in mid-2006, and 14 TFLOP/s in early 2004. Third-party programs for testing Mersenne numbers, such as Mlucas and Glucas, also, GIMPS reserves the right to change this EULA without notice and with reasonable retroactive effect. All Mersenne primes are in the form Mq, where q is the exponent, the prime number itself is 2q −1, so the smallest prime number in this table is 21398269 −1. Mn is the rank of the Mersenne prime based on its exponent, furthermore,71,027,647 is the largest exponent below which all other exponents have been tested at least once, so some Mersenne numbers between the 48th and the 49th have yet to be tested. ^ ‡ The number M74207281 has 22,338,618 decimal digits, to help visualize the size of this number, a standard word processor layout would require 5,957 pages to display it. If one were to print it out using standard printer paper, single-sided, whenever a possible prime is reported to the server, it is verified first before it is announced
2.
Logarithmic scale
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A logarithmic scale is a nonlinear scale used when there is a large range of quantities. Common uses include the strength, sound loudness, light intensity. It is based on orders of magnitude, rather than a linear scale. In particular our sense of hearing perceives equal ratios of frequencies as equal differences in pitch, the top left graph is linear in the X and Y axis, and the Y-axis ranges from 0 to 10. A base-10 log scale is used for the Y axis of the left graph. The top right graph uses a scale for just the X axis. A slide rule has logarithmic scales, and nomograms often employ logarithmic scales, the geometric mean of two numbers is midway between the numbers. Before the advent of graphics, logarithmic graph paper was a commonly used scientific tool. If both the vertical and horizontal axes of a plot are scaled logarithmically, the plot is referred to as a log–log plot, if only the ordinate or abscissa is scaled logarithmically, the plot is referred to as a semi-logarithmic plot. Bit Byte Decade John Napier Level Logarithm Logarithmic mean Preferred number Dehaene, Stanislas, Izard, Véronique, Spelke, Elizabeth, Pica, distinct intuitions of the number scale in Western and Amazonian indigene cultures. American Association for the Advancement of Science, why using logarithmic scale to display share prices
3.
Curve fitting
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Curve fitting is the process of constructing a curve, or mathematical function, that has the best fit to a series of data points, possibly subject to constraints. Curve fitting can involve either interpolation, where an exact fit to the data is required, or smoothing, in which a smooth function is constructed that approximately fits the data. A related topic is regression analysis, which focuses more on questions of statistical inference such as how much uncertainty is present in a curve that is fit to data observed with random errors. Fitted curves can be used as an aid for data visualization, to infer values of a function where no data are available, most commonly, one fits a function of the form y=f. Starting with a first degree polynomial equation, y = a x + b and this is a line with slope a. A line will connect any two points, so a first degree polynomial equation is an exact fit through any two points with distinct x coordinates. If the order of the equation is increased to a second degree polynomial and this will exactly fit a simple curve to three points. If the order of the equation is increased to a third degree polynomial and this will exactly fit four points. A more general statement would be to say it will exactly fit four constraints, each constraint can be a point, angle, or curvature. Angle and curvature constraints are most often added to the ends of a curve, identical end conditions are frequently used to ensure a smooth transition between polynomial curves contained within a single spline. Higher-order constraints, such as the change in the rate of curvature, many other combinations of constraints are possible for these and for higher order polynomial equations. If there are more than n +1 constraints, the curve can still be run through those constraints. An exact fit to all constraints is not certain, in general, however, some method is then needed to evaluate each approximation. The least squares method is one way to compare the deviations, there are several reasons given to get an approximate fit when it is possible to simply increase the degree of the polynomial equation and get an exact match. Even if a match exists, it does not necessarily follow that it can be readily discovered. Depending on the algorithm used there may be a divergent case and this situation might require an approximate solution. The effect of averaging out questionable data points in a sample, rather than distorting the curve to fit them exactly, runges phenomenon, high order polynomials can be highly oscillatory. If a curve runs through two points A and B, it would be expected that the curve would run somewhat near the midpoint of A and B, as well
4.
Euclid
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Euclid, sometimes called Euclid of Alexandria to distinguish him from Euclides of Megara, was a Greek mathematician, often referred to as the father of geometry. He was active in Alexandria during the reign of Ptolemy I, in the Elements, Euclid deduced the principles of what is now called Euclidean geometry from a small set of axioms. Euclid also wrote works on perspective, conic sections, spherical geometry, number theory, Euclid is the anglicized version of the Greek name Εὐκλείδης, which means renowned, glorious. Very few original references to Euclid survive, so little is known about his life, the date, place and circumstances of both his birth and death are unknown and may only be estimated roughly relative to other people mentioned with him. He is rarely mentioned by name by other Greek mathematicians from Archimedes onward, the few historical references to Euclid were written centuries after he lived by Proclus c.450 AD and Pappus of Alexandria c.320 AD. Proclus introduces Euclid only briefly in his Commentary on the Elements, Proclus later retells a story that, when Ptolemy I asked if there was a shorter path to learning geometry than Euclids Elements, Euclid replied there is no royal road to geometry. This anecdote is questionable since it is similar to a story told about Menaechmus, a detailed biography of Euclid is given by Arabian authors, mentioning, for example, a birth town of Tyre. This biography is generally believed to be completely fictitious, however, this hypothesis is not well accepted by scholars and there is little evidence in its favor. The only reference that historians rely on of Euclid having written the Elements was from Proclus, although best known for its geometric results, the Elements also includes number theory. The geometrical system described in the Elements was long known simply as geometry, today, however, that system is often referred to as Euclidean geometry to distinguish it from other so-called non-Euclidean geometries that mathematicians discovered in the 19th century. In addition to the Elements, at least five works of Euclid have survived to the present day and they follow the same logical structure as Elements, with definitions and proved propositions. Data deals with the nature and implications of information in geometrical problems. On Divisions of Figures, which only partially in Arabic translation. It is similar to a first-century AD work by Heron of Alexandria, catoptrics, which concerns the mathematical theory of mirrors, particularly the images formed in plane and spherical concave mirrors. The attribution is held to be anachronistic however by J J OConnor, phaenomena, a treatise on spherical astronomy, survives in Greek, it is quite similar to On the Moving Sphere by Autolycus of Pitane, who flourished around 310 BC. Optics is the earliest surviving Greek treatise on perspective, in its definitions Euclid follows the Platonic tradition that vision is caused by discrete rays which emanate from the eye. One important definition is the fourth, Things seen under a greater angle appear greater, proposition 45 is interesting, proving that for any two unequal magnitudes, there is a point from which the two appear equal. Other works are attributed to Euclid, but have been lost
5.
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
6.
Mersenne prime
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In mathematics, a Mersenne prime is a prime number that is one less than a power of two. That is, it is a number that can be written in the form Mn = 2n −1 for some integer n. They are named after Marin Mersenne, a French Minim friar, the first four Mersenne primes are 3,7,31, and 127. If n is a number then so is 2n −1. The definition is therefore unchanged when written Mp = 2p −1 where p is assumed prime, more generally, numbers of the form Mn = 2n −1 without the primality requirement are called Mersenne numbers. The smallest composite pernicious Mersenne number is 211 −1 =2047 =23 ×89, Mersenne primes Mp are also noteworthy due to their connection to perfect numbers. As of January 2016,49 Mersenne primes are known, the largest known prime number 274,207,281 −1 is a Mersenne prime. Since 1997, all newly found Mersenne primes have been discovered by the “Great Internet Mersenne Prime Search”, many fundamental questions about Mersenne primes remain unresolved. It is not even whether the set of Mersenne primes is finite or infinite. The Lenstra–Pomerance–Wagstaff conjecture asserts that there are infinitely many Mersenne primes,23 | M11,47 | M23,167 | M83,263 | M131,359 | M179,383 | M191,479 | M239, and 503 | M251. Since for these primes p, 2p +1 is congruent to 7 mod 8, so 2 is a quadratic residue mod 2p +1, since p is a prime, it must be p or 1. The first four Mersenne primes are M2 =3, M3 =7, M5 =31, a basic theorem about Mersenne numbers states that if Mp is prime, then the exponent p must also be prime. This follows from the identity 2 a b −1 = ⋅ = ⋅ and this rules out primality for Mersenne numbers with composite exponent, such as M4 =24 −1 =15 =3 ×5 = ×. Though the above examples might suggest that Mp is prime for all p, this is not the case. The evidence at hand does suggest that a randomly selected Mersenne number is more likely to be prime than an arbitrary randomly selected odd integer of similar size. Nonetheless, prime Mp appear to grow increasingly sparse as p increases, in fact, of the 2,270,720 prime numbers p up to 37,156,667, Mp is prime for only 45 of them. The lack of any simple test to determine whether a given Mersenne number is prime makes the search for Mersenne primes a difficult task, the Lucas–Lehmer primality test is an efficient primality test that greatly aids this task. The search for the largest known prime has somewhat of a cult following, consequently, a lot of computer power has been expended searching for new Mersenne primes, much of which is now done using distributed computing
7.
Fast Fourier transform
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A fast Fourier transform algorithm computes the discrete Fourier transform of a sequence, or its inverse. Fourier analysis converts a signal from its domain to a representation in the frequency domain. An FFT rapidly computes such transformations by factorizing the DFT matrix into a product of sparse factors. As a result, it manages to reduce the complexity of computing the DFT from O, which if one simply applies the definition of DFT, to O. Fast Fourier transforms are used for many applications in engineering, science. The basic ideas were popularized in 1965, but some algorithms had been derived as early as 1805, the DFT is obtained by decomposing a sequence of values into components of different frequencies. This operation is useful in many fields but computing it directly from the definition is too slow to be practical. The difference in speed can be enormous, especially for data sets where N may be in the thousands or millions. In practice, the time can be reduced by several orders of magnitude in such cases. The best-known FFT algorithms depend upon the factorization of N, but there are FFTs with O complexity for all N, even for prime N. Since the inverse DFT is the same as the DFT, but with the sign in the exponent. The development of fast algorithms for DFT can be traced to Gausss unpublished work in 1805 when he needed it to interpolate the orbit of asteroids Pallas and Juno from sample observations. His method was similar to the one published in 1965 by Cooley and Tukey. While Gausss work predated even Fouriers results in 1822, he did not analyze the computation time, between 1805 and 1965, some versions of FFT were published by other authors. Yates in 1932 published his version called interaction algorithm, which provided efficient computation of Hadamard, yates algorithm is still used in the field of statistical design and analysis of experiments. In 1942, Danielson and Lanczos published their version to compute DFT for x-ray crystallography, Cooley and Tukey published a more general version of FFT in 1965 that is applicable when N is composite and not necessarily a power of 2. To analyze the output of these sensors, a fast Fourier transform algorithm would be needed, garwin gave Tukeys idea to Cooley for implementation. Cooley and Tukey published the paper in a short six months
8.
Electronic Frontier Foundation
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The Electronic Frontier Foundation is an international non-profit digital rights group based in San Francisco, California. In April 1990, Barlow had been visited by a U. S. Federal Bureau of Investigation agent in relation to the theft, Barlow described the visit as complicated by fairly complete unfamiliarity with computer technology. I realized right away that before I could demonstrate my innocence, Barlow felt that his experience was symptomatic of a great paroxysm of governmental confusion during which everyones liberties would become at risk. Barlow posted an account of experience to The WELL online community and was contacted by Mitch Kapor. The pair agreed that there was a need to defend civil liberties on the Internet and this generated a large amount of publicity which led to offers of financial support from John Gilmore and Steve Wozniak. This generated further reaction and support for the ideas of Barlow, in late June, Barlow held a series of dinners in San Francisco with major figures in the computer industry to develop a coherent response to these perceived threats. Barlow considered that, The actions of the FBI and Secret Service were symptoms of a social crisis. America was entering the Information Age with neither laws nor metaphors for the appropriate protection, Barlow felt that to confront this a formal organization would be needed, he hired Cathy Cook as press coordinator, and began to set up what would become the Electronic Frontier Foundation. The Electronic Frontier Foundation was formally founded on July 10,1990, by Kapor and Barlow, who soon after elected Gilmore, Wozniak. Initial funding was provided by Kapor, Wozniak, and an anonymous benefactor, in 1990, Mike Godwin joined the organization as its first staff counsel. Then in 1991, Esther Dyson and Jerry Berman joined the EFF board of directors. C, the creation of the organization was motivated by the massive search and seizure on Steve Jackson Games executed by the United States Secret Service early in 1990. Similar but officially unconnected law-enforcement raids were being conducted across the United States at about time as part of a state–federal task force called Operation Sundevil. However, the Steve Jackson Games case, which became EFFs first high-profile case, was the rallying point around which EFF began promoting computer-. In 1993, their offices moved to 1001 G Street in Washington, more recently, the organization has been involved in defending Edward Felten, Jon Lech Johansen and Dmitry Sklyarov. The organization was located at Mitch Kapors Kapor Enterprises offices in Cambridge. By the fall of 1993, the main EFF offices were consolidated into an office, in Washington. During this time, some of EFFs attention focused on influencing national policy, in 1994, Berman parted ways with EFF and formed the Center for Democracy and Technology, while Drew Taubman briefly took the reins as executive director. There, it took up residence at John Gilmores Toad Hall
9.
Time (magazine)
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Time is an American weekly news magazine published in New York City. It was founded in 1923 and for decades was dominated by Henry Luce, a European edition is published in London and also covers the Middle East, Africa and, since 2003, Latin America. An Asian edition is based in Hong Kong, the South Pacific edition, which covers Australia, New Zealand and the Pacific Islands, is based in Sydney, Australia. In December 2008, Time discontinued publishing a Canadian advertiser edition, Time has the worlds largest circulation for a weekly news magazine, and has a readership of 26 million,20 million of which are based in the United States. As of 2012, it had a circulation of 3.3 million making it the eleventh most circulated magazine in the United States reception room circuit, as of 2015, its circulation was 3,036,602. Richard Stengel was the editor from May 2006 to October 2013. Nancy Gibbs has been the editor since October 2013. Time magazine was created in 1923 by Briton Hadden and Henry Luce, the two had previously worked together as chairman and managing editor respectively of the Yale Daily News. They first called the proposed magazine Facts and they wanted to emphasize brevity, so that a busy man could read it in an hour. They changed the name to Time and used the slogan Take Time–Its Brief and it set out to tell the news through people, and for many decades the magazines cover depicted a single person. More recently, Time has incorporated People of the Year issues which grew in popularity over the years, notable mentions of them were Barack Obama, Steve Jobs, Matej Turk, etc. The first issue of Time was published on March 3,1923, featuring Joseph G. Cannon, the retired Speaker of the House of Representatives, on its cover, a facsimile reprint of Issue No. 1, including all of the articles and advertisements contained in the original, was included with copies of the February 28,1938 issue as a commemoration of the magazines 15th anniversary. The cover price was 15¢ On Haddens death in 1929, Luce became the dominant man at Time, the Intimate History of a Publishing Enterprise 1923–1941. In 1929, Roy Larsen was also named a Time Inc. director, J. P. Morgan retained a certain control through two directorates and a share of stocks, both over Time and Fortune. Other shareholders were Brown Brothers W. A. Harriman & Co. the Intimate History of a Changing Enterprise 1957–1983. According to the September 10,1979 issue of The New York Times, after Time magazine began publishing its weekly issues in March 1923, Roy Larsen was able to increase its circulation by utilizing U. S. radio and movie theaters around the world. It often promoted both Time magazine and U. S. political and corporate interests, Larsen next arranged for a 30-minute radio program, The March of Time, to be broadcast over CBS, beginning on March 6,1931
10.
Pietro Cataldi
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Pietro Antonio Cataldi was an Italian mathematician. A citizen of Bologna, he taught mathematics and astronomy and also worked on military problems and his work included the development of continued fractions and a method for their representation. He was one of many mathematicians who attempted to prove Euclids fifth postulate, Cataldi discovered the sixth and seventh primes later to acquire the designation Mersenne primes by 1588. Although Cataldi also claimed that p=23,29,31 and 37 all also generate Mersenne primes, oConnor, John J. Robertson, Edmund F. Pietro Cataldi, MacTutor History of Mathematics archive, University of St Andrews
11.
Leonhard Euler
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He also introduced much of the modern mathematical terminology and notation, particularly for mathematical analysis, such as the notion of a mathematical function. He is also known for his work in mechanics, fluid dynamics, optics, astronomy, Euler was one of the most eminent mathematicians of the 18th century, and is held to be one of the greatest in history. He is also considered to be the most prolific mathematician of all time. His collected works fill 60 to 80 quarto volumes, more than anybody in the field and he spent most of his adult life in Saint Petersburg, Russia, and in Berlin, then the capital of Prussia. A statement attributed to Pierre-Simon Laplace expresses Eulers influence on mathematics, Read Euler, read Euler, Leonhard Euler was born on 15 April 1707, in Basel, Switzerland to Paul III Euler, a pastor of the Reformed Church, and Marguerite née Brucker, a pastors daughter. He had two sisters, Anna Maria and Maria Magdalena, and a younger brother Johann Heinrich. Soon after the birth of Leonhard, the Eulers moved from Basel to the town of Riehen, Paul Euler was a friend of the Bernoulli family, Johann Bernoulli was then regarded as Europes foremost mathematician, and would eventually be the most important influence on young Leonhard. Eulers formal education started in Basel, where he was sent to live with his maternal grandmother. In 1720, aged thirteen, he enrolled at the University of Basel, during that time, he was receiving Saturday afternoon lessons from Johann Bernoulli, who quickly discovered his new pupils incredible talent for mathematics. In 1726, Euler completed a dissertation on the propagation of sound with the title De Sono, at that time, he was unsuccessfully attempting to obtain a position at the University of Basel. In 1727, he first entered the Paris Academy Prize Problem competition, Pierre Bouguer, who became known as the father of naval architecture, won and Euler took second place. Euler later won this annual prize twelve times, around this time Johann Bernoullis two sons, Daniel and Nicolaus, were working at the Imperial Russian Academy of Sciences in Saint Petersburg. In November 1726 Euler eagerly accepted the offer, but delayed making the trip to Saint Petersburg while he applied for a physics professorship at the University of Basel. Euler arrived in Saint Petersburg on 17 May 1727 and he was promoted from his junior post in the medical department of the academy to a position in the mathematics department. He lodged with Daniel Bernoulli with whom he worked in close collaboration. Euler mastered Russian and settled life in Saint Petersburg. He also took on a job as a medic in the Russian Navy. The Academy at Saint Petersburg, established by Peter the Great, was intended to improve education in Russia, as a result, it was made especially attractive to foreign scholars like Euler
12.
2,147,483,647
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The number 2,147,483,647 is the eighth Mersenne prime, equal to 231 −1. It is one of four known double Mersenne primes. The primality of this number was proven by Leonhard Euler, who reported the proof in a letter to Daniel Bernoulli written in 1772, Euler used trial division, improving on Cataldis method, so that at most 372 divisions were needed. It thus improved upon the previous record-holding prime,6,700,417, also discovered by Euler, the number 2,147,483,647 remained the largest known prime until 1867. He repeated this prediction in his 1814 work A New Mathematical and Philosophical Dictionary, in fact a larger prime was discovered in 1855 by Thomas Clausen, though a proof was not provided. Furthermore,3,203,431,780,337 was proven to be prime in 1867, the number 2,147,483,647 is the maximum positive value for a 32-bit signed binary integer in computing. It is therefore the value for variables declared as integers in many programming languages. The appearance of the number often reflects an error, overflow condition, google later admitted that this was a joke. The data type time_t, used on operating systems such as Unix, is a signed integer counting the number of seconds since the start of the Unix epoch, and is often implemented as a 32-bit integer. The latest time that can be represented in this form is 03,14,07 UTC on Tuesday,19 January 2038 and this means that systems using a 32-bit time_t type are susceptible to the Year 2038 problem. Also, this number is in most browsers the highest to accept positive or negative z-index in Cascading Style Sheets, power of two Prime curios,2147483647
13.
Thomas Clausen (mathematician)
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For the Louisiana educator, see Thomas Clausen. Thomas Clausen was a Danish mathematician and astronomer and he eventually returned to Munich, where he conceived and published his best known works on mathematics. In 1842 Clausen was hired by the staff of the Tartu Observatory, works by Clausen include studies on the stability of Solar system, comet movement, ABC telegraph code and calculation of 250 decimals of Pi. In 1840 he discovered the Von Staudt–Clausen theorem, in 1854 he factored the sixth Fermat number as 264+1 =67280421310721 ×274177. Von Staudt–Clausen theorem Clausens formula Clausen function Biermann, Kurt-R, edward, Euler at 300, MAA Spectrum, Washington, DC, Math. America, pp. 217–225, ISBN 978-0-88385-565-2, MR2349552 Biography
14.
Computer Laboratory, University of Cambridge
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The Computer Laboratory is the computer science department of the University of Cambridge. As of 2007, it employs 35 academic staff,25 support staff,35 affiliated research staff, the current head of department is Professor Andy Hopper. The new laboratory was housed in the North Wing of the former Anatomy School, upon its foundation, it was intended to provide a computing service for general use, and to be a centre for the development of computational techniques in the University. The Cambridge Diploma in Computer Science was the world’s first postgraduate course in computing, starting in 1953. It inspired the world’s first business computer, LEO and it was replaced by EDSAC2, the first microcoded and bitsliced computer, in 1958. In 1961, David Hartley developed Autocode, one of the first high-level programming languages, also in that year, proposals for Titan, based on the Ferranti Atlas machine, were developed. Titan became fully operational in 1964 and EDSAC2 was retired the following year, in 1967, a full multi-user time-shared service for up to 64 users was inaugurated on Titan. In 2002, the Computer Laboratory launched the Cambridge Computer Lab Ring, the Computer Laboratory built and operated the world’s first fully operational practical stored program computer and offered the world’s first postgraduate taught course in computer science in 1953. It currently offers a 3-year undergraduate course and a 1-year masters course, members of the Computer Laboratory have been involved in the creation of many successful UK IT companies such as Acorn, ARM, nCipher and XenSource. A number of companies have been founded by staff and graduates and their names were featured in the new laboratory entrance in 2012. Some cited examples of companies are ARM, Autonomy, Aveva, CSR. One common factor they share is that key staff or founder members are drenched in university training, the Cambridge Computer Lab Ring was praised for its tireless work by Andy Hopper in 2012, at its tenth anniversary dinner. Ian Pratt Simon Crosby David L Tennenhouse Michael Burrows Andy Harter Andy Hopper
15.
Electronic delay storage automatic calculator
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Electronic delay storage automatic calculator was an early British computer. EDSAC was the electronic digital stored-program computer to go into regular service. Later the project was supported by J. Lyons & Co. Ltd. a British firm, work on EDSAC started during 1947, and it ran its first programs on 6 May 1949, when it calculated a table of squares and a list of prime numbers. EDSAC1 was finally shut down on 11 July 1958, having been superseded by EDSAC2, as soon as EDSAC was operational, it began serving the Universitys research needs. It used mercury delay lines for memory, and derated vacuum tubes for logic, cycle time was 1.5 ms for all ordinary instructions,6 ms for multiplication. Input was via five-hole punched tape and output was via a teleprinter, initially registers were limited to an accumulator and a multiplier register. In 1953, David Wheeler, returning from a stay at the University of Illinois, a magnetic tape drive was added in 1952 but never worked sufficiently well to be of real use. Until 1952, the main memory was only 512 18-bit words. The delay lines were arranged in two batteries providing 512 words each, the second battery came into operation in 1952. The full 1024-word delay line store was not available until 1955 or early 1956, the EDSACs main memory consisted of 1024 locations, though only 512 locations were initially installed. Each contained 18 bits, but the topmost bit was always due to timing problems. An instruction consisted of a five-bit instruction code, one bit, a ten bit operand. Numbers were either 17 bits or 35 bits long, unusually, the multiplier was designed to treat numbers as fixed-point fractions in the range −1 ≤ x <1, i. e. the binary point was immediately to the right of the sign. The accumulator could hold 71 bits, including the sign, allowing two long numbers to be multiplied without losing any precision, there was no division instruction and no way to directly load a number into the accumulator. There was no unconditional jump instruction, nor was there a call instruction - it had not yet been invented. The initial orders were hard-wired on a set of uniselector switches, by May 1949, the initial orders provided a primitive relocating assembler taking advantage of the mnemonic design described above, all in 31 words. This was the worlds first assembler, and arguably the start of the software industry. There is a simulation of EDSAC available and a description of the initial orders
16.
University of Central Missouri
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The University of Central Missouri, formerly Central Missouri State University, is a public state university located in Warrensburg, Missouri, United States. It serves more than 12,000 students from 49 states and 59 countries on its 1, UCM offers 150 programs of study, including 10 pre-professional programs,27 areas of teacher certification, and 37 graduate programs. Students also have the ability to study abroad in about 60 different countries in the world through the International Center, the University was founded in 1871 as Normal School No.2 and became known as Warrensburg Teachers College. The name was changed to Central Missouri State Teachers College in 1919, Central Missouri State College in 1945, in 1965, the institution established a graduate school. In 2006, the name was changed to the University of Central Missouri, there are 150 majors and minors,32 professional accreditations and 37 graduate programs. UCM has off-campus locations in Lees Summit, Missouri and provides online courses. College of Arts, Humanities, and Social Sciences, UCM students take College of Arts, Humanities, and Social Sciences courses that develop critical-thinking, writing, the current dean is Dr. Gersham Nelson. Accreditations include National Association of Schools of Music and National Council for Social Studies, U. S. News & World Report has cited UCM’s MBA program in America’s Best Graduate Schools. The current dean is Dr. Roger Best, other accreditations include Aviation Accreditation Board International and Council on Social Work Education. The current dean is Dr. Michael Wright, the college’s goal is to prepare students to be competent leaders in the rapidly changing global marketplace and to provide a high-quality work force for the future. The current dean is Dr. Alice Griefe, the Honors College, First-time incoming freshmen must have a minimum ACT score of 25 and a minimum cumulative high school GPA of 3.5 to be considered for admission to The Honors College. Once incoming freshmen have completed a semester at UCM as a student and have a college cumulative GPA of 3.5 or higher. Current UCM students or students transferring to UCM must have achieved a minimum cumulative college GPA of 3.5 or higher. Benefits of being an Honors College student include, but are not limited to, early enrollment, one-on-one advising with the Dean, smaller classes, Honors-only courses, the current dean of the Honors College and International Affairs is Dr. Joseph D. Lewandowski. The University of Central Missouri continues to hold an important role in the Great Internet Mersenne Prime Search, the GIMPS project at UCM is a university-wide effort managed by Dr. Curtis Cooper and Dr. Steven Boone. Centrals team is currently the No, the university has more than 200 student organizations with academic, cultural, recreational, community service and special interest clubs and associations. There are also more than 20 intramural sports to compete in, free movie nights on campus, freshman and sophomore students are required to live in one of the 16 residence halls their first year to help ease the adjustment from high school to college. Students can also choose to live in a Special Housing Interest Program, the University of Central Missouri is home to 26 Greek organizations, recruitment takes place in both the spring and fall semesters
17.
Megaprime
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A megaprime is a prime number with at least one million decimal digits. As of September 2016,208 megaprimes are known, including 192 definitely primes and 16 probable primes. The first to be found was the Mersenne prime 26972593−1 with 2,098,960 digits, discovered in 1999 by Nayan Hajratwala, the term bevaprime has been proposed as a term for a prime with at least 1,000,000,000 digits. In fact, almost all primes are megaprimes, as the amount of primes less than a million digits is finite. However, the vast majority of known primes are not megaprimes, entries labelled Prime have been proved prime, those labelled PRP have not. All numbers from 10999999 through 10999999 +593498 are known to be composite, and there is a high probability 10999999 +593499
18.
Time Inc.
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Time Inc. is an American mass media company founded on November 28,1922 by Henry Luce and Briton Hadden and based in New York City. It owns and publishes over 100 magazine brands, most notably its flagship Time, other magazines include Sports Illustrated, Travel + Leisure, Food & Wine, Fortune, People, InStyle, Life, Golf Magazine, Southern Living, Essence, Real Simple, and Entertainment Weekly. It also owns the UK magazine house Time Inc, UK, whose major titles include Whats on TV, NME, Country Life and Wallpaper. Time Inc. also operates over 60 websites and digital-only titles including MyRecipes, TheSnug, HelloGiggles, Time Inc. also owns the rights to LIFE, a well-known magazine that has been published in many different formats. Time Inc. currently owns and runs LIFE. com, a dedicated to news. In 1990, Time Inc. merged with Warner Communications to form the media conglomerate Time Warner and this merger lasted until the company was spun off on June 9,2014. Nightly discussions of the concept of a magazine led its founders Henry Luce and Briton Hadden. Later that same year, they formed Time Inc, having raised $86,000 of a $100,000 goal, the first issue of Time was published on March 3,1923, as the first weekly news magazine in the United States. Luce served as manager while Hadden was editor-in-chief. Luce and Hadden annually alternated year-to-year the titles of president and secretary-treasurer, upon Haddens sudden death in 1929, Luce assumed Haddens position. Luce launched the business magazine Fortune in February 1930 and created/founded the pictorial Life magazine in 1934 and he also produced The March of Time radio and newsreel series. By the mid 1960s, Time Inc. was the largest and most prestigious magazine publisher in the world, the main target was Luce, who had long opposed FDR. Historian Alan Brinkley argues the move was badly mistaken, for had Luce been allowed to travel, but stranded in New York City, Luces frustration and anger expressed itself in hard-edged partisanship. Luce, supported by Editor-in-Chief T. S. Matthews, appointed Whittaker Chambers as acting Foreign News editor in 1944, in 1963, recommendations from Time Inc. based on how it delivered magazines led to the introduction of ZIP codes by the United States Post Office. Luce, who remained editor-in-chief of all his publications until 1964, holding anti-communist sentiments, he used Time to support right-wing dictatorships in the name of fighting communism. The merger of Time Inc. and Warner Communications was announced on March 4,1989 and this caused Time to raise its bid for Warner to $14.9 billion in cash and stock. Paramount responded by filing a lawsuit in a Delaware court to block the Time/Warner merger, in 2008, Time Inc. launched Maghound, an internet-based magazine membership service that featured approximately 300 magazine titles from both Time Inc. brands and external publishing companies. On January 19,2010, Time Inc. acquired StyleFeeder, on March 6,2013, Time Warner announced plans to spin-off Time Inc. into a publicly traded company
19.
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
20.
Pythagorean prime
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A Pythagorean prime is a prime number of the form 4n +1. Pythagorean primes are exactly the odd numbers that are the sum of two squares. For instance, the number 5 is a Pythagorean prime, √5 is the hypotenuse of a triangle with legs 1 and 2. The first few Pythagorean primes are 5,13,17,29,37,41,53,61,73,89,97,101,109,113, by Dirichlets theorem on arithmetic progressions, this sequence is infinite. More strongly, for n, the numbers of Pythagorean and non-Pythagorean primes up to n are approximately equal. However, the number of Pythagorean primes up to n is frequently smaller than the number of non-Pythagorean primes. For example, the values of n up to 600000 for which there are more Pythagorean than non-Pythagorean odd primes are 26861 and 26862. Sum of one odd square and one square is congruent to 1 mod 4. Fermats theorem on sums of two states that the prime numbers that can be represented as sums of two squares are exactly 2 and the odd primes congruent to 1 mod 4. The representation of such number is unique, up to the ordering of the two squares. Another way to understand this representation as a sum of two squares involves Gaussian integers, the numbers whose real part and imaginary part are both integers. The norm of a Gaussian integer x + yi is the number x2 + y2, thus, the Pythagorean primes occur as norms of Gaussian integers, while other primes do not. Within the Gaussian integers, the Pythagorean primes are not considered to be prime numbers, similarly, their squares can be factored in a different way than their integer factorization, as p2 =22 =. The real and imaginary parts of the factors in these factorizations are the leg lengths of the right triangles having the given hypotenuses, in the finite field Z/p with p a Pythagorean prime, the polynomial equation x2 = −1 has two solutions. This may be expressed by saying that −1 is a quadratic residue mod p, in contrast, this equation has no solution in the finite fields Z/p where p is an odd prime but is not Pythagorean. Pythagorean Primes, including 5,13 and 137, sloanes A007350, Where prime race 4n-1 vs. 4n+1 changes leader. The On-Line Encyclopedia of Integer Sequences
21.
Pierpont prime
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A Pierpont prime is a prime number of the form 2 u 3 v +1 for some nonnegative integers u and v. That is, they are the prime numbers p for which p −1 is 3-smooth. They are named after the mathematician James Pierpont, who introduced them in the study of regular polygons that can be constructed using conic sections. It is possible to prove that if v =0 and u >0, then u must be a power of 2, if v is positive then u must also be positive, and the Pierpont prime is of the form 6k +1. Empirically, the Pierpont primes do not seem to be rare or sparsely distributed. There are 36 Pierpont primes less than 106,59 less than 109,151 less than 1020, there are few restrictions from algebraic factorisations on the Pierpont primes, so there are no requirements like the Mersenne prime condition that the exponent must be prime. As there are Θ numbers of the form in this range. Andrew M. Gleason made this explicit, conjecturing there are infinitely many Pierpont primes. According to Gleasons conjecture there are Θ Pierpont primes smaller than N, when 2 u >3 v, the primality of 2 u 3 v +1 can be tested by Proths theorem. As part of the ongoing search for factors of Fermat numbers. The following table gives values of m, k, and n such that k ⋅2 n +1 divides 22 m +1, the left-hand side is a Pierpont prime when k is a power of 3, the right-hand side is a Fermat number. As of 2017, the largest known Pierpont prime is 3 ×210829346 +1, whose primality was discovered by Sai Yik Tang, in the mathematics of paper folding, the Huzita axioms define six of the seven types of fold possible. It has been shown that these folds are sufficient to allow the construction of the points that solve any cubic equation. It follows that they allow any regular polygon of N sides to be formed, as long as N >3 and of the form 2m3nρ and this is the same class of regular polygons as those that can be constructed with a compass, straightedge, and angle-trisector. Regular polygons which can be constructed with compass and straightedge are the special case where n =0 and ρ is a product of distinct Fermat primes, themselves a subset of Pierpont primes. In 1895, James Pierpont studied the same class of regular polygons, Pierpont generalized compass and straightedge constructions in a different way, by adding the ability to draw conic sections whose coefficients come from previously constructed points. As he showed, the regular N-gons that can be constructed with these operations are the ones such that the totient of N is 3-smooth. Since the totient of a prime is formed by subtracting one from it, however, Pierpont did not describe the form of the composite numbers with 3-smooth totients. As Gleason later showed, these numbers are exactly the ones of the form 2m3nρ given above, the smallest prime that is not a Pierpont prime is 11, therefore, the hendecagon is the smallest regular polygon that cannot be constructed with compass, straightedge and angle trisector
22.
Pell number
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In mathematics, the Pell numbers are an infinite sequence of integers, known since ancient times, that comprise the denominators of the closest rational approximations to the square root of 2. This sequence of approximations begins 1/1, 3/2, 7/5, 17/12, and 41/29, so the sequence of Pell numbers begins with 1,2,5,12, and 29. The numerators of the sequence of approximations are half the companion Pell numbers or Pell–Lucas numbers, these numbers form a second infinite sequence that begins with 2,6,14,34. As with Pells equation, the name of the Pell numbers stems from Leonhard Eulers mistaken attribution of the equation, the Pell–Lucas numbers are also named after Édouard Lucas, who studied sequences defined by recurrences of this type, the Pell and companion Pell numbers are Lucas sequences. The Pell numbers are defined by the recurrence relation P n = {0 if n =0,1 if n =1,2 P n −1 + P n −2 otherwise. In words, the sequence of Pell numbers starts with 0 and 1, and then each Pell number is the sum of twice the previous Pell number and the Pell number before that. The first few terms of the sequence are 0,1,2,5,12,29,70,169,408,985,2378,5741,13860, …. The Pell numbers can also be expressed by the closed form formula P n = n − n 22, a third definition is possible, from the matrix formula = n. Pell numbers arise historically and most notably in the rational approximation to √2. If two large integers x and y form a solution to the Pell equation x 2 −2 y 2 = ±1 and that is, the solutions have the form P n −1 + P n P n. The approximation 2 ≈577408 of this type was known to Indian mathematicians in the third or fourth century B. C, the Greek mathematicians of the fifth century B. C. also knew of this sequence of approximations, Plato refers to the numerators as rational diameters. In the 2nd century CE Theon of Smyrna used the term the side and these approximations can be derived from the continued fraction expansion of 2,2 =1 +12 +12 +12 +12 +12 + ⋱. As Knuth describes, the fact that Pell numbers approximate √2 allows them to be used for accurate rational approximations to an octagon with vertex coordinates. All vertices are equally distant from the origin, and form uniform angles around the origin. Alternatively, the points, and form approximate octagons in which the vertices are equally distant from the origin. A Pell prime is a Pell number that is prime, the first few Pell primes are 2,5,29,5741, …. The indices of these primes within the sequence of all Pell numbers are 2,3,5,11,13,29,41,53,59,89,97,101,167,181,191, … These indices are all themselves prime. As with the Fibonacci numbers, a Pell number Pn can only be prime if n itself is prime, the only Pell numbers that are squares, cubes, or any higher power of an integer are 0,1, and 169 =132. However, despite having so few squares or other powers, Pell numbers have a connection to square triangular numbers