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Prime Numbers: Integers
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An integer greater than one is called a prime number if its only positive divisors are one and itself. For example, the number 10 is not prime because it is divisible by 2 and 5. A Mersenne prime is a prime of the form 2P-1. The study of Mersenne primes has been central to number theory since they were first discussed by Euclid in 350 BC. The man whose name they now bear, the French monk Marin Mersenne (1588-1648), made a famous conjecture on which values of p would yield a prime. It took 300 years and several important discoveries in mathematics to settle his conjecture.
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One method for determining whether a number is prime is to divide by all primes less than or equal to the square root of that number. If any of the divisions come out as an integer, then the original number is not a prime. Otherwise, it is a prime. One need not actually calculate the square root; once one sees that the quotient is less than the divisor, one can stop. This is known as trial division; it is the simplest primality test and it quickly becomes impractical for testing large integers because the number of possible factors grows exponentially as the number of digits in the number-to-be-tested increases.
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The Prime Numbers are one of the most interesting topics in mathematics. Primes appear to be the building blocks of multiplication—you can express all positive integers as a unique set of primes factors—but there is not a simple equation to generate a list of primes, nor is there a simple way to determine if any given number is prime.
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Precisely defined yet enticingly elusive, prime numbers occupy a central place in number theory. Evenly divisible only by themselves and 1, these special integers—2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, and so on—pose all sorts of conundrums.
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In ring theory, a branch of abstract algebra, the term “prime element” has a specific meaning. Here, a non-zero, non-unit ring element a is defined to be prime if whenever a divides bc for ring elements b and c, then a divides at least one of b or c. With this meaning, the additive inverse of any prime number is ... prime. In other words, when considering the set of integers as a ring, −7 is a prime element. Without further specification, however, “prime number” always means a positive integer prime. Among rings of complex algebraic integers, Eisenstein primes, and Gaussian primes may also be of interest.
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This means that information about prime or not prime only needs to occupy 8 bits. A ten byte file would hold prime or not prime for 300 integers. It works because N*30 has 2,3,5 as factors, no matter what the value of N. (N*30)+3 has 3 as a factor since 3 is ... a factor of 30. (N*30)+(x*3) also has 3 as a factor since it could be written as (N*10 + x)*3.
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Integers