RE: [PrimeNumbers] How many primes?
> From: Nathan Russell [mailto:nrussell@...]Back to basics - remember than when dealing with the infinite, you must use
> 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?
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
(*) PNT: pi(x) ~ x/ln(x) ==>
(proportion of primes <= x) ~ 1/ln(x)
Therefore given epsilon, choose N>exp(100/epsilon)
(Who hasn't done an epsilon-delta proof for 25 years!)
Virus checked by MessageLabs Virus Control Centre.