As mentioned in Terry Tao's comment to [this question](http://mathoverflow.net/q/44466/5810), it is constructively known
<br>
that there are primes between sufficiently large cubes. $\:$ [According to wikipedia](https://en.wikipedia.org/wiki/Prime_gap#Upper_bounds),
<br>
"there exists a constant $\: \theta < 1 \:$ such that $\;\;\; \pi \hspace{-0.03 in}\left(x\hspace{-0.04 in}+\hspace{-0.04 in}x^{\hspace{.02 in}\theta}\right)-\pi \hspace{.03 in}(x) \: \sim \: \dfrac{x^{\hspace{.02 in}\theta}}{\log(x)} \;\;\;$ as $x$ tends to infinity".

Is there any somewhat-similar corresponding effective result for the density of primes in short intervals?

Motivation:<br>
Such a result could yield an effective randomized reduction from subset sum to [this variant of factoring](http://cstheory.stackexchange.com/q/4769/6973).