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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
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