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?



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 > ∞. 


If one wants an explicit bound on N_{0}, apparently this can be gleaned from a Ph.D. thesis by Pierre Dusart (in French) which contains the result that for all x > 3275 there is a prime between x and x(1 + 1/(2 ln^{2} x)). So we can take N_{0} to be max(3275,exp((2 epsilon)^{−1/2})). 

