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Prime Numbers: Factors
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Prime numbers belong to an exclusive world of intellectual conceptions. They are one of those marvelous notions that enjoy simple, elegant descriptions, yet lead to extreme complexity in the details. The basic notion of primality can be accessible to a child, yet no human mind harbors anything like a complete picture. In this book the authors concentrate on the computational aspect of finding and characterizing primes, but will often digress into the theoretical domain in order to illuminate, justify, and underscore the practical import of the computational algorithms. The book will be an indispensable reference for professionals interested in prime numbers and encryption, cryptography, factoring algorithms, elliptic curve arithmetic, and many more computational issues related to primes and factoring. Readers can test their understanding at the end of each chapter with a variety of exercises ranging from very easy to extremely difficult.
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After the Greeks, little happened with the study of prime numbers until the 17th century. In 1640 Pierre de Fermat stated (without proof) Fermat's little theorem (later proved by Leibnitz and Euler). A special case of Fermat's theorem may have been known much earlier by the Chinese. Fermat conjectured that all numbers of the form 22n + 1 are prime (they are called Fermat numbers) and he verified this up to n = 4. However, the very next Fermat number 232+1 is composite (one of its prime factors is 641), as Euler discovered later, and in fact no further Fermat numbers are known to be prime. The French monk Marin Mersenne looked at primes of the form 2p - 1, with p a prime.
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[One] way of creating numbers out of prime factor configuration patterns would be to treat the two classes (as represented by 12 and 18) the same. Such a system would feature fewer distinct patterns. The difference between the two systems is very close to the difference between permutations and combinations.
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There is a well-known formula that generates a "perfect" number from a Mersenne prime. A perfect number is one whose factors add up to the number itself. The smallest perfect number is 6 = 1 + 2 + 3. The newly discovered perfect number is 2^6972592 * (2^6972593-1). This number is 4,197,919 digits long!
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A mnemonic device is useful in working with larger numbers and their prime factor configuration patterns and numbers. The idea is to associate each configuration not so much with its pfcpn itself, but rather with the smallest number that exhibits that pfcpn. This is what the chart immediately above does.
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There are two types of SAT* questions concerning prime numbers that you need to be familiar with. One type concerns factorization, which will be covered later. The other deals with the properties of smaller prime numbers. The above example is that type of problem.
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