1.
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

2.
Personal computer
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A personal computer is a multi-purpose electronic computer whose size, capabilities, and price make it feasible for individual use. PCs are intended to be operated directly by a end-user, rather than by an expert or technician. In the 2010s, PCs are typically connected to the Internet, allowing access to the World Wide Web, personal computers may be connected to a local area network, either by a cable or a wireless connection. In the 2010s, a PC may be, a multi-component desktop computer, designed for use in a location a laptop computer, designed for easy portability or a tablet computer. In the 2010s, PCs run using a system, such as Microsoft Windows, Linux. The very earliest microcomputers, equipped with a front panel, required hand-loading of a program to load programs from external storage. Before long, automatic booting from permanent read-only memory became universal, in the 2010s, users have access to a wide range of commercial software, free software and free and open-source software, which are provided in ready-to-run or ready-to-compile form. Since the early 1990s, Microsoft operating systems and Intel hardware have dominated much of the computer market, first with MS-DOS. Alternatives to Microsofts Windows operating systems occupy a minority share of the industry and these include Apples OS X and free open-source Unix-like operating systems such as Linux and Berkeley Software Distribution. Advanced Micro Devices provides the alternative to Intels processors. PC is an initialism for personal computer, some PCs, including the OLPC XOs, are equipped with x86 or x64 processors but not designed to run Microsoft Windows. PC is used in contrast with Mac, an Apple Macintosh computer and this sense of the word is used in the Get a Mac advertisement campaign that ran between 2006 and 2009, as well as its rival, Im a PC campaign, that appeared in 2008. Since Apples transition to Intel processors starting 2005, all Macintosh computers are now PCs, the “brain” may one day come down to our level and help with our income-tax and book-keeping calculations. But this is speculation and there is no sign of it so far, in the history of computing there were many examples of computers designed to be used by one person, as opposed to terminals connected to mainframe computers. Using the narrow definition of operated by one person, the first personal computer was the ENIAC which became operational in 1946 and it did not meet further definitions of affordable or easy to use. An example of an early single-user computer was the LGP-30, created in 1956 by Stan Frankel and used for science and it came with a retail price of $47, 000—equivalent to about $414,000 today. Introduced at the 1965 New York Worlds Fair, the Programma 101 was a programmable calculator described in advertisements as a desktop computer. It was manufactured by the Italian company Olivetti and invented by the Italian engineer Pier Giorgio Perotto, the Soviet MIR series of computers was developed from 1965 to 1969 in a group headed by Victor Glushkov

3.
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