October 27th, 2013, 09:36 PM  #21  
Math Team Joined: Mar 2012 From: India, West Bengal Posts: 3,871 Thanks: 86 Math Focus: Number Theory  Re: The set of primes as an algebraic structure Quote:
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October 27th, 2013, 09:47 PM  #22  
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: The set of primes as an algebraic structure Quote:
 
October 27th, 2013, 09:54 PM  #23  
Math Team Joined: Mar 2012 From: India, West Bengal Posts: 3,871 Thanks: 86 Math Focus: Number Theory  Re: The set of primes as an algebraic structure Quote:
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October 27th, 2013, 10:29 PM  #24 
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: The set of primes as an algebraic structure
That should be enough to define the LagariasOdlyzko analytic method then, right? So we have pi(x). The inverse logarithmic integral is similarly easy to define with those operators (plus simple ones like + and *), so just use li^1 and two iterations of pi(x) to get a (very) close estimate of the nth prime. Probably with more work you could get an exact formula (in addition to being asymptotically efficient) but I won't stick my head out trying to come up with that tonight. 
October 28th, 2013, 07:46 AM  #25  
Senior Member Joined: Aug 2012 Posts: 2,356 Thanks: 738  Re: The set of primes as an algebraic structure Quote:
http://en.wikipedia.org/wiki/Formula_for_primes The article's only about formulas that generate primes ... doesn't say anything about specifically generating the nth prime. There are some interesting formulas in the article. For example if RH is true then there's a number A such that floor(A^3^n) is prime for all positive integers n. The number A is called Mills' constant. http://en.wikipedia.org/wiki/Mills%27_constant But the sequence skips a lot of primes.  
October 28th, 2013, 10:09 AM  #26 
Senior Member Joined: Feb 2012 Posts: 628 Thanks: 1  Re: The set of primes as an algebraic structure
Back on topic, I came up with a binary operation that is closed under the primes but does not form a group: Define a % b to be the largest prime factor of the sum of a + b. Some examples: 3 % 5 = 2 2 % 5 = 7 11 % 7 = 3 13 % 7 = 5 2 % 13 = 5 5 % 11 = 2 There is no identity element and even if you tried to force one of the primes to be an identity element by definition there would not be unique inverses. For example, I could arbitrarily decide that 2 % x = x % 2 = 2 by definition, and then define a % b in the normal way when 2 was not involved. But then you would have 3 % 5 = 11 % 5 = 2 so 5 would not have a unique inverse. But I was thinking that it would make the most sense for 2 to be the identity element regardless of what the binary operation was. The rationale for this is that 2 is the smallest prime and also the only one which is even. 
October 28th, 2013, 11:29 AM  #27  
Math Team Joined: Mar 2012 From: India, West Bengal Posts: 3,871 Thanks: 86 Math Focus: Number Theory  Re: The set of primes as an algebraic structure Quote:
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October 28th, 2013, 12:33 PM  #28  
Senior Member Joined: Aug 2012 Posts: 2,356 Thanks: 738  Re: The set of primes as an algebraic structure Quote:
To be clear: I'm under the impression that the only way to compute pi(n) exactly is to count the primes onebyone. You can get an approximation to pi(n) for large n with logarithms, but it's not exact.  
October 28th, 2013, 01:13 PM  #29 
Senior Member Joined: Feb 2012 Posts: 628 Thanks: 1  Re: The set of primes as an algebraic structure
You are correct. There is no known explicit formula for . Also, there is no known explicit formula for the nth prime. Formulas like the one mathbalarka posted are similar to in that they are very simple yet still cannot be expressed explicitly.

October 28th, 2013, 01:34 PM  #30  
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: The set of primes as an algebraic structure Quote:
 

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