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

*To*: mathgroup at smc.vnet.net*Subject*: [mg88587] 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 03:24:28 -0400 (EDT)*References*: <fvpcpn$mge$1@smc.vnet.net> <200805080814.EAA14338@smc.vnet.net>

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>