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Re: Re: Question on the Limiting Value of Ratios of Consecuative Primes...
*To*: mathgroup at smc.vnet.net
*Subject*: [mg88591] 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 03:25:12 -0400 (EDT)
*References*: <fvpcpn$mge$1@smc.vnet.net> <200805080814.EAA14338@smc.vnet.net> <8DE86472-F622-4298-A54F-2172ADC748B0@mimuw.edu.pl>
I forgot to add that the question of whether there are infinitely many
twin primes or not still remains unsolved (http://mathworld.wolfram.com/TwinPrimes.html
). Cramer's conjecture that I mentioned in my post also seems not to
have been proved.
But the proof given based on Montgomery's result that I sketched
below is a really a proof, and does not depen on any unproved
conjectures.
Andrzej Kozlowski
On 9 May 2008, at 06:46, Andrzej Kozlowski wrote:
>
> 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
>>
>
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