Unconditional result for the existence of primes in intervals of length one

Question

Dusart proved in 2010 that there's at least one prime between $x$ and $\left(1 + \frac{1}{25\ln^2x}\right)x$ for $x \geq 396738$

My question is: Is Dusart's the smallest known interval with at least one prime?

Secondly, Is there is an unconditional result for the existence of primes in intervals of lenght one, i.e., given $a>0$ and $b>0$, such that $b-a=1$, then find conditions on $a$ and $b$ in which there exist a prime $p$ verifying $a<p<b$.

Answer 1

The asymptotically smallest known interval which always contains a prime for sufficiently large $x$ is $[x, x + x^{0.525 + o(1)}]$, by a result of Baker, Harman and Pintz.

This is considerably smaller than Dusart, but it is somewhat inexplicit. I am not sure if someone has done the work to extract a concrete, explicit constant from their paper, although it should be possible in principle.

Conjecturally, we believe that the smallest interval is more like $[x,x + O((\log x)^{2+o(1)})]$, which would be exponentially smaller still.

Your second question makes very little sense. The only time there is a prime in the open interval $(a,a+1)$ is when $a$ is a non-integer that rounds up to a prime. It would be impossible to give a more generic condition than that.

Answer 2

The prime number theorem implies that for any $\epsilon > 0$, there exists an integer $N$ such that for all $x > N$, there exists a prime between $x$ and $x(1 + \epsilon)$.