Let $\varepsilon$ be an arbitrary small positive number. Can we prove that there exist an $n\in Z$ such that the interval $[2^n,(1+\varepsilon)2^n]$ contain a prime number?

Yes, by any of several results (starting with Hoheisel) which assert that for sufficiently large $x$ one has at least one prime in $(x,x+x^\theta)$, where $\theta$ is given in advance and is some real number less than one. The current record is like 0.535 for $\theta$.
– The Masked AvengerJan 31 '14 at 7:42

7

Actually all you need is the prime number theorem, which guarantees primes in $(x,(1+\epsilon)x)$ if $x$ is large. So for all large $n$ there are primes in your interval.
– LuciaJan 31 '14 at 9:46