Re: Re: Question on the Limiting Value of Ratios of Consecuative Primes...

*To*: mathgroup at smc.vnet.net*Subject*: [mg88591] Re: [mg88555] Re: Question on the Limiting Value of Ratios of Consecuative Primes...*From*: Andrzej Kozlowski <akoz at mimuw.edu.pl>*Date*: Fri, 9 May 2008 03:25:12 -0400 (EDT)*References*: <fvpcpn$mge$1@smc.vnet.net> <200805080814.EAA14338@smc.vnet.net> <8DE86472-F622-4298-A54F-2172ADC748B0@mimuw.edu.pl>

I forgot to add that the question of whether there are infinitely many twin primes or not still remains unsolved (http://mathworld.wolfram.com/TwinPrimes.html ). Cramer's conjecture that I mentioned in my post also seems not to have been proved. But the proof given based on Montgomery's result that I sketched below is a really a proof, and does not depen on any unproved conjectures. Andrzej Kozlowski On 9 May 2008, at 06:46, Andrzej Kozlowski wrote: > > On 8 May 2008, at 17:14, David W.Cantrell wrote: > >> "Richard Palmer" <rhpalmer at gmail.com> wrote: >>> Is there some analytic limit to the ratio of consecuative primes? >> >> Yes. The limit is 1. >> >> Since there are infinitely many twin primes, it's obvious that, if >> the >> limit exists, it must be 1. >> >> I don't know how to show nicely that the limit exists. > > This result follows from the following one proved in H.L. > Montgomery "Topics in Multiplicative Number Theory" (Springer 1971): > > For any epsilon >0 and x > x0(epsilon), there is a prime in the > interval [x,x+x^(3/5 + epsilon)]. > > This means that for arbirarily small epsilon and a sufficiently > large Prime[n], the ratio Prime[n+1]/Prime[n] is less than (Prime[n] > +Prime[n]^(3/5+epsilon))/Prime[n] = 1 + Prime[n]^(epsilon-2/5), > which can be made arbitrarily close to 1 (for epsilon <2/5). > > Actually, there has been a lot of interest in the long term > behaviour of the difference of consecutive primes. Montgomery > mentions the following which he attributes to Cramer: > > Limit[(Prime[n+1]-Prime[n])/Log[Prime[n]]^2,n->Infinity] == 1 > > (this is not meant to be a Mathematica formula, since Mathematica > does not have the notion of the limit of a sequence but only a > mathematical statement written in the Mathematica notation) > > but I am not sure if that is a theorem or only a conjecture. > > Andrzej Kozlowski > > >> >> >>> The expression Limit[Prime[i]/Prime[i+1],{i,->Infinity}] returns >>> unevaluated. >> >> I suspect that is just as well. My understanding is that >> Mathematica is not >> designed to deal with limits of sequences and that, had you gotten an >> answer, it should not have been trusted. >> >>> Plotting Table[ Prime[i]/Prime[i+1],{i,1,20000}] shows a lot >>> of structure with a minimum of 3/5. >> >> I suppose I see the "structure" to which you refer: various strings >> of >> points which could be visualized as lying on smooth curves. The >> points >> which form the uppermost string correspond to the ratios of the twin >> primes. The points which form the next string down correspond to >> the ratios >> of primes having a difference of 4. And then the points which form >> the next >> string down correspond to the ratios of primes having a difference >> of 6. >> Etc. >> >> David W. Cantrell >> >

**References**:**Re: Question on the Limiting Value of Ratios of Consecuative Primes...***From:*"David W.Cantrell" <DWCantrell@sigmaxi.net>