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

  • To: mathgroup at smc.vnet.net
  • Subject: [mg88523] Re: Question on the Limiting Value of Ratios of Consecuative Primes...
  • From: Jean-Marc Gulliet <jeanmarc.gulliet at gmail.com>
  • Date: Wed, 7 May 2008 07:09:09 -0400 (EDT)
  • Organization: The Open University, Milton Keynes, UK
  • References: <fvpcpn$mge$1@smc.vnet.net>

Richard Palmer 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.

You could use something like the expression below to explore numerically 
the limit on a wider range of prime numbers. (Table build a list, i.e. 
consume more and more memory, whereas For will just iterate and change 
only one value when required.)

In[142]:= Timing@
  Module[{rat, lowest = 1},
   For[i = 1, i <= 10^7, i++, rat = Prime[i]/Prime[i + 1];
    If[rat < lowest, lowest = rat]]; lowest]

Out[142]= {139.046, 3/5}

Regards,
-- Jean-Marc




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