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Prime Numbers: Integers
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Prime numbers have long fascinated amateur and professional mathematicians. An integer greater than one is called a prime number if its only divisors are one and itself. The first prime numbers are 2, 3, 5, 7, 11, etc. 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 first Mersenne primes are 3, 7, 31, 127 (corresponding to P = 2, 3, 5, 7).
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Prime numbers have fascinated mathematicians, both professional and amateur, for centuries. They are often called the "building blocks" of the set of positive integers because every positive integer either is a prime number itself or can be written as the product of prime numbers in exactly one way. Few objects in mathematics are as simple to describe, yet few have so much depth and mystery to their structure. These numbers (2, 3, 5, 7, 11, ...) show up seemingly at random, yet some beautiful patterns underlie their distribution. This chapter explores some of these patterns and connects the prime numbers to some of the deepest problems in all of mathematics.
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Prime numbers are numbers that are only divisible by themselves and one. Ancient Greek mathematicians first studied them. Euclid proved that there are infinite prime numbers. In the "Fundamental Theorem of Arithmetic", Euclid prooved that every integer can be written as a product of primes.
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Prime numbers lie at the core of some of the oldest and most perplexing questions in mathematics. Evenly divisible only by themselves and 1, they are the building blocks of integers. In recent decades, prime numbers have emerged from their starring roles in mathematical research (SN: 5/25/02, p. 324: Available to subscribers at http://www.sciencenews.org/20020525/fob4.asp) by becoming prized commodities—as elements in a cryptographic scheme widely used to keep digital messages secret (SN: 2/6/99, p. 95).
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![[An image of some prime numbers]](http://retrieverimages.lycos.com/images/p/r/i/prime-numbers/i/001.gif)
Prime numbers are special because they are the elementary building blocks of the multiplicative structure on the integers; every integer can be written in only one way as a product of its prime factors. The mathematically precise version of this assertion is known as the The Fundamental Theorem of Arithmetic.
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In Book IX of the Elements, Euclid proves that there are infinitely many prime numbers. This is one of the first proofs known which uses the method of contradiction to establish a result. Euclid ... gives a proof of the Fundamental Theorem of Arithmetic: Every integer can be written as a product of primes in an essentially unique way.
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