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

• To: mathgroup at smc.vnet.net
• Subject: [mg88555] Re: Question on the Limiting Value of Ratios of Consecuative Primes...
• From: "David W.Cantrell" <DWCantrell at sigmaxi.net>
• Date: Thu, 8 May 2008 04:14:25 -0400 (EDT)
• References: <fvpcpn\$mge\$1@smc.vnet.net>

```"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.

> 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

```

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