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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?

Tito Piezas III
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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 $x > p_2$.)

Tito Piezas III
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  • Great! That's what I've been looking for! Thanks so much. –  Aug 30 '15 at 14:16
  • So Dusart didn't know about this result? Dusart's paper is from 2010, this one from 2002... –  Aug 30 '15 at 14:17
  • @hetajr: If it was his thesis, he must have known. – Tito Piezas III Aug 30 '15 at 14:18
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    I don't really see in which sense this result is stronger than Dusart's. This one gives an interval of length $\mathrm{const}\cdot x$, while Dusart's result gives length $o(x)$... – user2097 Aug 30 '15 at 14:33
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    To be precise, the quoted result is only better for $x \lesssim 10^{462}$. For larger $x$ Dusart's result is better. – Winther Aug 30 '15 at 16:05
  • I was hoping someone would point that out. – DanielWainfleet Aug 30 '15 at 16:10
  • @Winther: It did occur to me that Dusart's result might eventually overtake Theorem 3. Serves me right for not exploring the implications thoroughly. :) – Tito Piezas III Aug 30 '15 at 16:49
  • @TitoPiezasIII: Dear Tito: Is the Theorem 3 you mentioned related to the Rieman hypothesis. The same question is for the Dusart boundes described in the question. – DER Apr 11 '18 at 14:37
  • @Winther: Is the Theorem 3 you mentioned related to the Rieman hypothesis. The same question is for the Dusart boundes described in the question. – DER Apr 11 '18 at 14:43
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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 $x/\log^2 x$.

Erick Wong
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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.

Ravi Fernando
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