1.
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 By the early century, it had been superseded by number theory.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

2.
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

3.
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

4.
Primecoin
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Primecoin is a peer-to-peer open source cryptocurrency that implements a unique scientific computing proof-of-work system. Primecoins proof-of-work system searches for chains of prime numbers, Primecoin was created by a person or group of people who use the pseudonym Sunny King. This entity is also related with the cryptocurrency Peercoin, the Primecoin source code is copyrighted by a person or group called “Primecoin Developers”, and distributed under a conditional MIT/X11 software license. Primecoin has been described as the cause of spot shortages of dedicated servers because at the time it was only possible to mine the currency with CPUs. For the same reason, Primecoin used to be the target of malware writers, there has been some attempts at approximating this number. Difficulty adjustment is more frequent Primecoin protocol adjusts its difficulty slightly after every block, the difficulty change that occurs each block is targeted at achieving target of one new block created once per minute. As comparison the bitcoin protocol adjusts its difficulty every 2016 blocks, faster transaction confirmations Since Primecoin blocks are generated 8 to 10 times as fast as bitcoin blocks on average, Primecoin transactions are confirmed approximately 8 to 10 times as fast. Primecoin uses the finding of prime chains composed of Cunningham chains and bi-twin chains for proof-of-work, the system is designed so that the work is efficiently verifiable by all nodes on the Primecoin network. To meet this requirement, the size of the numbers in the system cannot be too large. The Primecoin proof-of-work system has the characteristics, Primecoins work takes the form of prime number chains. Finding the prime number chains becomes exponentially harder as the length is increased. Verification of the reasonably sized prime number chains can be performed efficiently by all network nodes, mersenne primes are precluded due to their extremely large size. Three types of prime number chains are accepted as proof-of-work, Cunningham chain of the first kind, Cunningham chain of the second kind. Other cryptocurrencies including bitcoin commonly use a Hashcash type of proof-of-work based on SHA-256 hash calculations, list of largest known Cunningham chains of given length includes several results generated by Primecoin miners. Anarcho-capitalism Anonymous Internet banking Alternative currency Crypto-anarchism Money supply Moores law Prime number Cunningham chain Official website

5.
Primorial
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The rest of this article uses the latter interpretation. The name primorial, coined by Harvey Dubner, draws an analogy to primes the same way the name relates to factors. For the nth prime number pn, the primorial pn# is defined as the product of the first n primes, p n # ≡ ∏ k =1 n p k, where pk is the kth prime number. For instance, p5# signifies the product of the first 5 primes, the first six primorials pn# are,1,2,6,30,210,2310. The sequence also includes p0# =1 as empty product, asymptotically, primorials pn# grow according to, p n # = e n log n, where o is little-o notation. This is equivalent to, n # = {1 if n =0,1 # × n if n is prime # if n is composite. For example, 12# represents the product of those primes ≤12,12 # =2 ×3 ×5 ×7 ×11 =2310, since π =5, this can be calculated as,12 # = p π # = p 5 # =2310. Consider the first 12 primorials n#,1,2,6,6,30,30,210,210,210,210,2310,2310. We see that for composite n every term n# simply duplicates the preceding term #, in the above example we have 12# = p5# = 11# since 12 is a composite number. The natural logarithm of n# is the first Chebyshev function, written ϑ or θ, primorials n# grow according to, ln ∼ n. The idea of multiplying all known primes occurs in some proofs of the infinitude of the prime numbers, primorials play a role in the search for prime numbers in additive arithmetic progressions. For instance, 7009223613394100000♠2236133941 + 23# results in a prime, beginning a sequence of thirteen primes found by repeatedly adding 23#, 23# is also the common difference in arithmetic progressions of fifteen and sixteen primes. Every highly composite number is a product of primorials, primorials are all square-free integers, and each one has more distinct prime factors than any number smaller than it. For each primorial n, the fraction φ/n is smaller than it for any lesser integer, any completely multiplicative function is defined by its values at primorials, since it is defined by its values at primes, which can be recovered by division of adjacent values. Base systems corresponding to primorials have a proportion of repeating fractions than any smaller base. Every primorial is a sparsely totient number, the n-compositorial of a composite number n is the product of all composite numbers up to and including n. The n-compositorial is equal to the n-factorial divided by the primorial n#, the compositorials are 1,4,24,192,1728, 7004172800000000000♠17280, 7005207360000000000♠207360, 7006290304000000000♠2903040, 7007435456000000000♠43545600, 7008696729600000000♠696729600