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

*To*: mathgroup at smc.vnet.net*Subject*: [mg88595] Re: [mg88467] Question on the Limiting Value of Ratios of Consecuative Primes...*From*: DrMajorBob <drmajorbob at att.net>*Date*: Fri, 9 May 2008 03:25:56 -0400 (EDT)*References*: <29833150.1210083473580.JavaMail.root@m08>*Reply-to*: drmajorbob at longhorns.com

Here's a sketchy proof that the limit is one, based on the Prime Number Theorem: prime[primePi_] = First@Quiet[x /. Solve[primePi == x/Log[x], x]] -primePi ProductLog[-(1/primePi)] Limit[prime[i]/prime[i + 1], i -> Infinity] 1 I understand that Mathematica's Prime function works by inverting PrimePi; I've inverted, instead, an asymptotic approximation of PrimePi. Bobby On Tue, 06 May 2008 05:38:53 -0500, Richard Palmer <rhpalmer at gmail.com> wrote: > Is there some analytic limit to the ratio of consecuative primes? The > expression Limit[Prime[i]/Prime[i+1],{i,->Infinity}] returns > unevaluated. > Plotting Table[ Prime[i]/Prime[i+1],{i,1,20000}] shows a lot of structure > with a minimum of 3/5. > -- DrMajorBob at longhorns.com