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RE: [PrimeNumbers] How many primes?

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  • Paul Jobling
    ... Back to basics - remember than when dealing with the infinite, you must use limits... Due to the PNT, if you give me a percentage, epsilon (say 1%), I can
    Message 1 of 9 , May 29 1:42 AM
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      > From: Nathan Russell [mailto:nrussell@...]
      > I thought the primes could be mapped onto the natural
      > numbers, since there
      > are Aleph-null of each - wouldn't this mean the numbers are
      > equal, and the
      > precentage is either 100 or undefined?

      Back to basics - remember than when dealing with the infinite, you must use
      limits...

      Due to the PNT, if you give me a percentage, epsilon (say 1%), I can give you
      a number N such that the proportion of primes between 1 and any M>=N is less
      than epsilon (*). Since you can make epsilon as small as you like, the limit
      is 0%.

      (*) PNT: pi(x) ~ x/ln(x) ==>
      (proportion of primes <= x) ~ 1/ln(x)
      Therefore given epsilon, choose N>exp(100/epsilon)

      Regards,

      Paul

      (Who hasn't done an epsilon-delta proof for 25 years!)


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