       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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