LYCOS RETRIEVER 
Prime Numbers
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The oldest known proof for the statement that there are infinitely many prime numbers is given by the Greek mathematician Euclid in his Elements (Book IX, Proposition 20). Euclid states the result as "there are more than any given [finite] number of primes", and his proof is essentially the following:
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The Twin Prime conjecture states that there are infinitely many such pairs. No one has been able to verify whether it's really true. Results on the gaps between successive primes (not just twin primes) announced this past spring will perhaps provide insight to proving the twin prime conjecture.
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To find a prime number is easy using the test. Pick a number of the magnitude you'd like the prime number to have. Test the number using 20 something iterations. If the test says "No", subtract or add 1 to your number and do the test again. Repeat until prime number "probably" found.
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Many numbers occur in nature, and inevitably some of these are prime. There are... relatively few examples of numbers that appear in nature because they are prime. For example, most starfish have 5 arms, and 5 is a prime number. However there is no evidence to suggest that starfish have 5 arms because 5 is a prime number. Indeed, some starfish have different numbers of arms. Echinaster luzonicus normally has six arms, Luidia senegalensis has nine arms, and Solaster endeca can have as many as twenty arms.
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Rather than trying to find out if a particular number is prime, it is quicker to find out which numbers are composites. The prime numbers are everything left over. The most popular way to discover prime numbers is with prime sieve. The most famous prime sieve is the Sieve of Eratosthenes. This sieve lists numbers in sets of six or twelve so that it is easy to remove common multiples. The Sieve of Eratosthenes was popular because you can perform it easily on a chalkboard or piece of paper.
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This document contains a bunch of programs to figure out prime numbers. The first rendition requires 1:23 (one minute 23 seconds) to calculate all primes below ten million. From there, improvements are made, dead ends are explored and backed out of, leading to the final version, which finds all primes below 40
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Prime Numbers