1.
Digital electronic computer
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In computer science, a digital electronic computer is a computer machine which is both an electronic computer and a digital computer. Examples of a digital electronic computers include the IBM PC, the Apple Macintosh as well as modern smartphones, an example of a computer which is both non-digital and non-electronic is the ancient Antikythera mechanism found in Greece. All kinds of computers, whether they are digital or analog, as of 2014, most digital electronic supercomputers are also cluster computers, a technology that can be used at home in the form of small Beowulf clusters. Parallel computation is possible with non-digital or non-electronic computers. An example of a computation system using the abacus would be a group of human computers using a number of abacus machines for computation. A digital computer can perform its operations in the system, in binary. As of 2014, all electronic computers commonly used, whether personal computers or supercomputers, are working in the binary number system. A few ternary computers using ternary logic were built mainly in the Soviet Union as research projects, a digital electronic computer is not necessarily a transistorized computer, before the advent of the transistor, computers used vacuum tubes. One such possible development is the memristor, advances in quantum computing, DNA computing, optical computing or other technologies could lead to the development of more powerful computers in the future. Digital computers are inherently best described by mathematics, while analog computers are most commonly associated with continuous mathematics. The philosophy of digital physics views the universe as being digital, konrad Zuse wrote a book known as Rechnender Raum in which he described the whole universe as one all-encompassing computer. Abacus ENIAC EDVAC List of vacuum tube computers History of computing hardware List of transistorized computers

2.
Lehmer sieve
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Lehmer sieves are mechanical devices that implement sieves in number theory. Lehmer sieves are named for Derrick Norman Lehmer and his son Derrick Henry Lehmer, the father was a professor of mathematics at the University of California, Berkeley at the time, and his son followed in his footsteps as a number theorist and professor at Berkeley. A sieve in general is intended to find the numbers which are remainders when a set of numbers are divided by a second set, generally, they are used in finding solutions of diophantine equations or to factor numbers. A Lehmer sieve will signal that such solutions are found in a variety of ways depending on the particular construction, the first Lehmer sieve in 1926 was made using bicycle chains of varying length, with rods at appropriate points in the chains. As the chains turned, the rods would close electrical switches, and when all the switches were closed simultaneously, creating an electrical circuit. Lehmer sieves were very fast, in one particular case factoring 293 +1 =3 ×3 ×529510939 ×715827883 ×2903110321 in 3 seconds, built in 1932, a device using gears was shown at the Century of Progress Exposition in Chicago. These had gears representing numbers, just as the chains had before, holes left open were the remainders sought. When the holes lined up, a light at one end of the device shone on a photocell at the other and this incarnation allowed checking of five thousand combinations a second. In 1936, a version was built using 16 mm film instead of chains, brushes against the rollers would make electrical contact when the hole reached the top. Again, a sequence of holes created a complete circuit. Several Lehmer sieves are on display at the Computer History Museum, since then, the same basic idea has been used to design sieves in integrated circuits or software. Sieve of Eratosthenes Lehmer, D. N, hunting big game in the theory of numbers, Scripta Mathematica,1, 229–235. The mechanical combination of forms, American Mathematical Monthly, Mathematical Association of America,35, 114–121, doi,10. 2307/2299504. Also online at the Antique Computer home page, beiler, Albert H. Recreations in the Theory of Numbers, Dover, chap. XX, XXI. Lehmer Sieves, by Dr. Michael R. Williams, Head Curator of The Computer History Museum The Computer History Museum page about Lehmer Sieves

3.
Number theory
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Number theory or, in older usage, arithmetic is a branch of pure mathematics devoted primarily to the study of the integers. It is sometimes called The Queen of Mathematics because of its place in the discipline. Number theorists study prime numbers as well as the properties of objects out of integers or defined as generalizations of the integers. Integers can be considered either in themselves or as solutions to equations, questions in number theory are often best understood through the study of analytical objects that encode properties of the integers, primes or other number-theoretic objects in some fashion. One may also study real numbers in relation to rational numbers, the older term for number theory is arithmetic. By the early century, it had been superseded by number theory. The use of the arithmetic for number theory regained some ground in the second half of the 20th century. In particular, arithmetical is preferred as an adjective to number-theoretic. The first historical find of a nature is a fragment of a table. The triples are too many and too large to have been obtained by brute force, the heading over the first column reads, The takiltum of the diagonal which has been subtracted such that the width. The tables layout suggests that it was constructed by means of what amounts, in language, to the identity 2 +1 =2. If some other method was used, the triples were first constructed and then reordered by c / a, presumably for use as a table. It is not known what these applications may have been, or whether there could have any, Babylonian astronomy, for example. It has been suggested instead that the table was a source of examples for school problems. While Babylonian number theory—or what survives of Babylonian mathematics that can be called thus—consists of this single, striking fragment, late Neoplatonic sources state that Pythagoras learned mathematics from the Babylonians. Much earlier sources state that Thales and Pythagoras traveled and studied in Egypt, Euclid IX 21—34 is very probably Pythagorean, it is very simple material, but it is all that is needed to prove that 2 is irrational. Pythagorean mystics gave great importance to the odd and the even, the discovery that 2 is irrational is credited to the early Pythagoreans. This forced a distinction between numbers, on the one hand, and lengths and proportions, on the other hand, the Pythagorean tradition spoke also of so-called polygonal or figurate numbers