khaled Posted September 15, 2011 Share Posted September 15, 2011 Lets say that we have the set [math]P = { 2, 3, 5, 7, ... }[/math], where [math]P[/math] is the set of all prime numbers ... The series [math]P_1 + P_2 + P_3 + ...[/math] does it converge or diverge ..? [math]\sum_{P_i \in P} P_i \; = \; ?[/math] Link to comment Share on other sites More sharing options...

Shadow Posted September 15, 2011 Share Posted September 15, 2011 Think. If that fails, plot. Link to comment Share on other sites More sharing options...

khaled Posted September 15, 2011 Author Share Posted September 15, 2011 figure: Spiral-Plot of prime numbers under 2026 I think I can't get it, I'll do the plots later ... Link to comment Share on other sites More sharing options...

baric Posted September 15, 2011 Share Posted September 15, 2011 Primes are an infinite set of positive integers, and zero is not prime. The series diverges. Link to comment Share on other sites More sharing options...

DrRocket Posted September 15, 2011 Share Posted September 15, 2011 The series diverges. rapidly Link to comment Share on other sites More sharing options...

khaled Posted September 16, 2011 Author Share Posted September 16, 2011 (edited) The series diverges. rapidly Is that enough to say: [math]\sum_{P_i \in P} P_i = \infty[/math] ..? Because, besides Euclid's work, all formula that can be used to generate prime numbers create series that diverges ... Edited September 16, 2011 by khaled Link to comment Share on other sites More sharing options...

Shadow Posted September 16, 2011 Share Posted September 16, 2011 figure: Spiral-Plot of prime numbers under 2026 I think I can't get it, I'll do the plots later ... What you posted is Sacks spiral, the same as Ulams spiral except using an Archimedean spiral instead of a square one, and has nothing whatsoever to do with sums of primes. In any case, what I meant by plot was [math]f(x) = \sum_{i=1}^{x} P_{i}[/math]. Link to comment Share on other sites More sharing options...

DrRocket Posted September 16, 2011 Share Posted September 16, 2011 Is that enough to say: [math]\sum_{P_i \in P} P_i = \infty[/math] ..? Because, besides Euclid's work, all formula that can be used to generate prime numbers create series that diverges ... That is to say that [math] \lim_{n \to \infty} P_n = \infty [/math] so that to think that the sum might actually converge is absurd. You are considering an "infinite sum" of positive integers, with the individual terms growing rapidly -- which is obvious from the fact that there are infinitely many primes and more so from the prime number theorem. Link to comment Share on other sites More sharing options...

baric Posted September 16, 2011 Share Posted September 16, 2011 That is to say that [math] \lim_{n \to \infty} P_n = \infty [/math] so that to think that the sum might actually converge is absurd. I agree and was wondering if the original poster was simply confused about the meaning of converging and diverging. Link to comment Share on other sites More sharing options...

forumreader Posted September 19, 2011 Share Posted September 19, 2011 i got it Link to comment Share on other sites More sharing options...

Johnny_Pencilface Posted September 23, 2011 Share Posted September 23, 2011 Furthermore, is there an equation than can express the nth value of any prime number? For example, where n=1, F(n)=2 etc. Link to comment Share on other sites More sharing options...

imatfaal Posted September 23, 2011 Share Posted September 23, 2011 Nope Link to comment Share on other sites More sharing options...

DrRocket Posted September 23, 2011 Share Posted September 23, 2011 Furthermore, is there an equation than can express the nth value of any prime number? For example, where n=1, F(n)=2 etc. No In fact no simple expression that produces only prime numbers is known, let alone one that would list all of them in increasing order. Link to comment Share on other sites More sharing options...

baric Posted September 23, 2011 Share Posted September 23, 2011 No In fact no simple expression that produces only prime numbers is known, let alone one that would list all of them in increasing order. That is probably the Holy Grail of Number Theory, and one that we all suspect is an impossibility. Link to comment Share on other sites More sharing options...

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