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December 8th, 2008, 09:58 AM  #1 
Senior Member Joined: Nov 2007 Posts: 258 Thanks: 0  A new proof of the infinitude of primes
I discovered this proof using the basic idea of the sieve of Eratosthenes. Enjoy! http://monkeytex.bradcater.webfactional ... f/?uid=470 
December 8th, 2008, 12:41 PM  #2 
Global Moderator Joined: Nov 2006 From: UTC 5 Posts: 16,046 Thanks: 938 Math Focus: Number theory, computational mathematics, combinatorics, FOM, symbolic logic, TCS, algorithms  Re: A new proof of the infinitude of primes
The proof is correct, good job.

December 8th, 2008, 04:10 PM  #3 
Newbie Joined: Dec 2008 Posts: 1 Thanks: 0  Re: A new proof of the infinitude of primes
Hi Bruno. "And since the second term is nonzero, sn < 1 for all n, and hence there are innitely many primes." If you run your argument for all n, you are sieving with an infinite number of primes p_n, which is what you're trying to prove. 
December 8th, 2008, 04:32 PM  #4 
Senior Member Joined: Nov 2007 Posts: 258 Thanks: 0  Re: A new proof of the infinitude of primes
Hello Sakura! Well the idea is that for any finite n, s_n < 1. Hence there is at least one number which has not been crossed out; i.e. one number which is not divisible by the primes p_1, ..., p_n. Obviously s_n converges to 1, but it is never 1. I'm not exactly sure what you are trying to point out! 
December 10th, 2008, 07:13 AM  #5  
Global Moderator Joined: Nov 2006 From: UTC 5 Posts: 16,046 Thanks: 938 Math Focus: Number theory, computational mathematics, combinatorics, FOM, symbolic logic, TCS, algorithms  Re: A new proof of the infinitude of primes Quote:
In essence: * Start with 2. It is a prime which divides half the integers. * 2 doesn't divide the odd numbers, so amongst the odd numbers there is at least one prime. Choose it to be p. * The set {2, p} divides only (1  1/2)(1  1/p) of the primes, so amongst the numbers relatively prime to 2 and p there is at least one prime. Choose it to be q. * The set (2, p, q}... This uses the Axiom of Choice.  
December 11th, 2008, 07:14 PM  #6 
Senior Member Joined: Oct 2008 Posts: 215 Thanks: 0  Re: A new proof of the infinitude of primes
We could use the idea to prove the infinitude of primes. But I think we'd better using the following approach. Assumming there're only finite primes, and all of them are Let Let T to be the union of natural number divisible by one of . It is easy to calculate the number of integers in intersection of and T. 
December 12th, 2008, 07:57 AM  #7 
Global Moderator Joined: Nov 2006 From: UTC 5 Posts: 16,046 Thanks: 938 Math Focus: Number theory, computational mathematics, combinatorics, FOM, symbolic logic, TCS, algorithms  Re: A new proof of the infinitude of primes
That variation actually proves an upper bound on p_n  though not a very good one. 

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