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Question about density of prime numbers?

  1. Sep 24, 2013 #1
    It is known that prime numbers become sparser and sparser, with the average distance between one prime number and the next increasing as n approaches infinity. Dividing an even number by 2 results in a bottom half from 1 to n / 2 and a top half from n / 2 to n. For a particular sufficiently large n, would the top half from n / 2 to n no longer contains any prime numbers?
     
    Last edited: Sep 24, 2013
  2. jcsd
  3. Sep 24, 2013 #2
    No, there is always a prime between n and 2n (which is the same thing as between n/2 and n). This is called Bertrand's Postulate, but despite the name, it's a theorem.

    In fact, there are a LOT of primes in that interval, and the bigger n is, the more there are. The density of the primes near n is like ##{1 \over \log n}##, so a rough estimate for the number of primes between n and 2n is ##{n \over \log 2n} \approx {n \over \log n}##.
     
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