# Strong Bertrand postulate

Is it known that for every epsilon there is N_0 such that all intervals of the form [N, (1+\epsilon)*N], where N > N_0, contain prime numbers?

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This follows from the Prime Number Theorem. Let π(n) be the number of primes less than n. Then π(n) ~ n/log(n); it follows π((1+ε)n)-π(n) -> ∞ as n -> ∞.

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How string is the statement that π(n) ~ n/log(n)? One certainly imagines it to be strong enough to say something specific about pi(n)-pi(n-1), but this is certainly not the typical Prime Number Theorem people study. –  Ilya Nikokoshev Oct 26 '09 at 23:55
~ has a precise definition: f(n) ~ g(n) means that lim_{n -> infty} f(n)/g(n)=1. That's strong enough to derive the desired conclusion. –  David Speyer Oct 26 '09 at 23:59
Mm, yes it is so. –  Ilya Nikokoshev Oct 27 '09 at 0:04