1. Limited time only! Sign up for a free 30min personal tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

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}##.
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook