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Is there a constant $\alpha$ such that:

$$P_{n+1} < P_n \cdot \left(\frac{n+1}{n}\right)^\alpha$$

Or

$$\lim_{n\to\infty}\frac{\ln\frac{P_{n+1}}{P_n}}{\ln\frac{n+1}{n}} < +\infty$$

Where $P_n$ is $n$-th prime number.

In the table The 80 known maximal prime gaps:

  • $\alpha < 33.3$ with $P_n=1693182318746371$

  • $\alpha < 35.77$ with $P_n=18361375334787046697$

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As $P_n$ is asymptotically $n\log n$, your question is equivalent to the following. Is it true that $$P_{n+1}-P_n\ll\log n?$$ In other words, is it true that the actual gap between primes is always at most a constant times the average (expected) gap? The answer is "no" by a 1931 result of Westzynthius. For the best known lower bound (for infinitely many prime gaps) see the paper of Ford-Green-Konyagin-Maynard-Tao.

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  • $\begingroup$ Thanks you very much $\endgroup$ – Đào Thanh Oai Sep 28 '19 at 16:23

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