## RE: [PrimeNumbers] How many primes?

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• ... 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
> 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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