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Infinite Number of Primes


Date: 04/14/98 at 15:44:14
From: Domenico Simonetti
Subject: Numbers not divisible

I'm looking for the greatest number not divisible calculated by some 
supercomputer or university's department or some mathematician, and 
want to know if they have or not discovered whether these numbers are 
or are not infinite.

Can you help me please? Thank you very much!


Date: 04/14/98 at 15:59:51
From: Doctor Kate
Subject: Re: Numbers not divisible

By "numbers not divisible," do you mean prime numbers?  Numbers which 
are only divisible by 1 and by themselves (like 2, 3, 5, 7, 11, 13, 
17...)?  If so, then the largest one recently found is about 2 million 
digits long or something astronomical like that. You can read about 
The Largest Known Primes on the Web:

  http://www.utm.edu/research/primes/largest.html   

Primes *are* in fact infinite, and the proof was found a long time 
ago. It's a beautiful proof, so even if you *aren't* asking about 
primes, I'm going to tell you this gorgeous proof:

Assume that primes are not infinite (i.e. there is a finite number of 
primes). We will come to a contradiction.

If there is a finite number of primes, then we can multiply them all 
together. This will be a number divisible by every single prime. Now 
add 1. We now have a number that, when you divide it by any prime, 
will have a remainder of 1 (make sure you believe this).

This number is a problem, because it can't be prime, and it isn't 
divisible by any prime. So what is it? All composites (divisible 
numbers) are divisible by some primes, so it can't be prime or 
composite. Therefore this number can't exist. This is a contradiction, 
so our original assumption (that prime numbers are finite) is wrong.

Thus prime numbers are infinite in number.

I suggest you look up primes in the Dr. Math FAQ for more information 
if you are interested.

   http://mathforum.org/dr.math/faq/faq.prime.num.html   

Hope this helps.

-Doctor Kate,  The Math Forum
Check out our web site! http://mathforum.org/dr.math/   
    
Associated Topics:
Middle School Number Sense/About Numbers

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