8

So far, I've gathered these intervals that contain at least one prime but are there other important results that I might have overlooked?

1. Oppermann's Conjecture: $[x^2,x^2+x]$

2. Bertrand's postulate: $[x,2x]$

3. Fourier optimization and prime gaps: $[x, x + \frac{22}{25}\sqrt{x} \log(x)]$

4. Riemann Hypothesis: $[x, x + x^{1/2 + \epsilon}]$

5. Baker, Harman, and Pintz: $[x,x+x^{0.525}]$

6. Cramér's conjecture: $[x, x + C \log^2(x)]$

7. Dudek: $[x^3,(x+1)^3]$

8. Legendre's conjecture: $[x^2,(x+1)^2]$

9. Dusart: $[\left(1 + \frac{1}{25\ln^2x}\right)x,x]$

10. Short effective intervals containing primes: $]x\big(1-\tfrac{1}{28314000}\big),\;x]$

vengy
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    It's important to note that not all of these are proved! #1,3,6,8 are open problems (and #4 is a conditional result on RH). – Greg Martin Oct 24 '23 at 19:35
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    And they hold only for sufficiently large $x$, which is known in some cases explicitly. For 10., it is $>10726905041$. For 7. it is $x≥\exp(\exp(33.217)) $ and so on. – Dietrich Burde Oct 24 '23 at 20:28
  • in 6. numerical evidence suggests $C=1$ works for $x \geq 8$ https://en.wikipedia.org/wiki/Prime_gap#Numerical_results – Will Jagy Oct 24 '23 at 22:16
  • in the small table of large "merit," the condition $C \leq 1$ is confirmed for those numbers $(M,g)$ when $M^2 < g,$ because $M = \frac{g}{\log p}$ – Will Jagy Oct 24 '23 at 22:27
  • Could you please add a link to each example (wikipedia or wolfram). – Gevorg Hmayakyan Nov 06 '23 at 04:19

1 Answers1

2

Here is one more.

The Firoozbakht conjecture is equivalent to the statement that there is a prime in the interval

$$ [x,x + \ln^2x - \ln x + 1] $$

This is stronger than the Cramer-Granville heuristics and the conjecture is believed to the false but has not been disproved.

Nilotpal Sinha
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