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Prime Numbers: Factors
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Prime numbers in the tables are indicated by the letter P. If a number is not prime, the table gives the smallest factor by which the number is divisible. All even numbers are omitted from the tables because none, except 2, are primes and all have a factor of 2.
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Much of elementary number theory was developed while deciding how to handle large numbers, how to characterize their factors and discover those which are prime. In short, the tradition of seeking large primes (especially the Mersennes) has been long and fruitful It is a tradition well worth continuing.
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The reason why 1 is said not to be a prime number is merely convenience. For example, if 1 was prime then the prime factorization of 6 would not be unique since 2 times 3 = 1 times 2 times 3. A number that can be written as a product of prime numbers is composite. Thus there are three types of natural numbers: primes, composites, and 1.
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If p is a prime number other than 2 and 5, 1/p is always a recurring decimal, whose period is p − 1 or a divisor of p − 1. This can be deduced directly from Fermat's little theorem. 1/p expressed likewise in base q (other than base 10) has similar effect, provided that p is not a prime factor of q. The article on recurring decimals shows some of the interesting properties.
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Until the 19th century, most mathematicians considered the number 1 a prime, and there is still a large body of mathematical work that is valid despite labeling 1 a prime, such as the work of Stern and Zeisel. Henri Lebesgue is said to be the last professional mathematician to call 1 prime. The change in label occurred so that it can be said that “each number has a unique factorization into primesâ€[2][3][4] (this is the fundamental theorem of arithmetic, which under the historical primality definition, is not valid as stated).
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Note that the definition of a prime number doesn't allow 1 to be a prime number: 1 only has one factor, namely 1. Prime numbers have exactly two factors, not "at most two" or anything like that. When a number has more than two factors it is called a composite number.
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