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

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  • 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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