LYCOS RETRIEVER 
Prime Numbers: Product
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If you were wanting all of the prime numbers instead of just those up to some arbitrary limit, then you are out of luck. There are an infinite number of primes. A greek named Euclid proved this a couple thousand years ago. He proved this by assuming that there were a finite number of primes and prime N was the biggest prime. Then, one could multiply all of the primes together and add one to the product. This new number is not divisible by any of the primes that were multiplied together, therefore it is either a prime itself, or it is a product of a prime larger than the one we earlier assumed to be the largest prime.
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Prime numbers, such as 17 and 23, are those that can only be divided by themselves and one. They are the most important objects in mathematics because, as the ancient Greeks discovered, they are the building blocks of all numbers—any of which can be broken down into a product of primes. (For example, 105 = 3 x 5 x 7.) They are the hydrogen and oxygen of the world of mathematics, the atoms of arithmetic. They ... represent one of the greatest challenges in mathematics.
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Suppose that there are only a finite number of prime numbers. Let m be the least common multiple of all of them. (This least common multiple was ... considered in proposition IX.14. It wasn't noted in the proof of that proposition that the least common multiple is the product of the primes, and it isn't noted in this proof, either.)
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WALNUT CREEK, Calif., Sept. 19 /PRNewswire/ -- Skyler Technology, Inc., today officially unveiled its Prime Processing technology and real-time data processing engine at DEMOfall 2005, the premier conference for new product and technology launches. Skyler's real-time data processing engine for the financial services industry allows users to thoroughly analyze trade data at unprecedented levels of speed, compression, and sophistication.
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The proof is sometimes phrased in a way that leads the student to conclude that P + 1 must itself be prime, and think that Euclid's proof says the prime product plus 1 is always prime. The simple example of (2 · 3 · 5 · 7 · 11 · 13) + 1 = 30,031 = 59 · 509 (both primes) shows that this is not the case. In fact, any set of primes which does not include 2 will have an odd product. Adding 1 to this product will always produce an even number, which will be divisible by 2 (and therefore not be prime).
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