By Tang X.

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**Sample text**

058 But for all of this steady fall in the ratio p(N)/N, the primes never peter out completely. Euclid's proof of this fact remains to this day a marvelous example of logical elegance. Here it is. The idea is to demonstrate that if you start to list the primes as a sequence p1, p2, p3, . . , this list continues forever. To prove this, you show that if you have listed all the primes up to some prime pn, then you can always find another prime to add to the list: the list never stops. Euclid's ingenious idea was to look at the number P = p1 × p2 × .

That's the first item of information. Next, I tell you that the first domino is knocked over. In terms of our abbreviation, I tell you that P(1)is true. That's the second item of information. On the basis of those two pieces of information, you can clearly conclude that the entire row of dominoes will fall down. 11). Of course, in real life, the row of dominoes will be finite. But exactly the same idea will work in a more abstract setting, where the abbreviation P(n) refers to some other event, one that makes sense for every natural number n, not just a domino falling down.

To make sure the computation was correct, Clarkson asked veteran prime hunter David Slowinski to check the result. A previous holder of a number of record prime discoveries, Slowinski, who works for Cray Research, used a Cray T90 supercomputer to repeat the calculation. Clarkson is one of over four thousand volunteers who use all their spare computer time to search for Mersenne primes as members of GIMPS, the Great Internet Mersenne Prime Search. GIMPS is a worldwide project coordinated by George Woltman, a programmer living in Orlando, Florida, who wrote and supplies the software.