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### Euclid's Proof on the Infinitude of Primes

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Date: 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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Associated Topics:
High School Number Theory

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