Euclid's Proof on the Infinitude of PrimesDate: 10/31/95 at 22:24:46 From: Anonymous Subject: Prime numbers Dear Dr. Math, I have been asked the question: "Which Greek mathematician proved that there is no greatest prime number?" Would you be able to help me with this problem? Thanks a lot. David Goldstein Date: 10/31/95 at 23:30:44 From: Doctor Sarah Subject: Re: Prime numbers Hello there - If you have Web access, check out http://www.utm.edu/research/primes/largest.html for Euclid's Proof of the Infinitude of Primes. It has lots of information on big primes, and about Euclid says: "Kummer's restatement of Euclid's proof is as follows: Suppose that there exist only finitely many primes p1 < p2 < ... < pr. Let N = (p1)(p2)...(pr) > 2. The integer N-1, being a product of primes, has a prime divisor pi in common with N; so, pi divides N - (N-1) =1, which is absurd! Quoted from page 4 of Ribenboim's 'Book of Prime Number Records'. Ribenboim's book contains "nine and a half" proofs of this theorem!" -Doctor Sarah, The Geometry Forum |
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