What's the smallest known interval containing at least one prime number? Wikipedia says that Dusart proved in 2010 that there's at least one prime between $x$ and $\left(1 + \frac{1}{25\ln^2x}\right)x$ for $x \geq 396738$. For $x_0 = 396738$, this implies a prime between $x_0$ and $x_0+96$.
My question is: Is Dusart's the smallest known interval with at least one prime? Or maybe someone obtained better results? 
 A: Actually, there is a better result below a bound (see Winther's comment) known since 2003. See "Short effective intervals containing primes" by Ramaré and Saouter. In page 13, 
Theorem 3: "Let $x>10726905041$. Then the interval,
$$\Big]x\big(1-\tfrac{1}{28314000}\big),\;x\Big]$$
contains at least one prime."
For example, if we plug in $x_0 = 10726905042$, Theorem 3 states that there is at least one prime between,
$$x_0-378.8\quad \text{and}\quad x_0\tag1$$
However, if we use Dusart's 2010 result, then the interval is,
$$x_0\quad \text{and}\quad x_0+804377.8\tag2$$
So within that bound, Ramaré and Saouter's theorem give a much shorter interval than Dusart's.
(Edit: Note that the gap between the two consecutive primes, 
$$p_2-p_1 = 10726905041-10726904659 = 382 > 378.8$$ 
and is #36 of the first 75 maximal gaps. Thus, a reason why $x > p_2$.)
A: The best unconditionally known result is the one listed below Dusart's on the page you linked to (https://en.wikipedia.org/wiki/Bertrand%27s_postulate#Better_results):  Baker, Harman, and Pintz proved in 2001 (link to paper) that there is always a prime in the interval $[x - x^{0.525}, x]$ for $x$ sufficiently large.  (They don't specify how large is sufficiently large, but assert that this could be determined "with enough effort".)  For reasonably large $x$, this $x^{0.525}$ gap is much smaller than either $x/28314000$ or $x/(25 \ln^2 x)$.
Additionally, Harald Cramér proved conditionally on the Riemann hypothesis that the $x^{0.525}$ can be reduced to $O(\sqrt x \ln x)$, and he conjectured (https://en.wikipedia.org/wiki/Cram%C3%A9r%27s_conjecture) that it can be reduced even to $O(\ln^2 x)$.  This seems very far beyond what we could hope to prove in the near future, but it's the strongest bound that looks likely to be true.
A: Asymptotically, Baker-Harman-Pintz is the best known result: for sufficiently large $x$ there is always a prime in the interval $[x,x+x^{0.525}]$, but I'm not sure that anyone's worked out an explicit value for "sufficiently large".  This paper of Dudek gives an explicit estimate for $[x^3, (x+1)^3]$, which is again asymptotically much better than $cx$ or $x/\log^2 x$.
