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In this post we denote for integers $n\geq 1$ the $n$-th Ramanujan prime as $R_n$ (thus the sequence A104272 from the On-Line Encyclopedia of Integer Sequences), I add a conjecture that I think can be potentially interesting due that Ramanujan primes are related to the prime-counting function $\pi(x)$ in their definition. It is about if does possible to get a similar expression than the known as Andrica's conjecture. There are some known facts in the literature about the sequence or Ramanujan primes, for example the Wikipedia Ramanujan prime.

Conjecture 1. We've that $\sqrt{R_{n+1}}-\sqrt{R_{n}}<1$ for all integer $n>42$.

Question. Is it possible to prove or refute previous conjecture? Add feedback for previous inequality, even in the case that the problem is very difficut to solve. Many thanks.

My opinion is that this conjecture is interesting and is at research level (I believe that it is interesting in the context of prime gaps, and that will be difficult to elucidate their veracity). I wrote programs in Pari/GP to check the first few hundred of the terms of this sequence to know what about of the mentioned inequality.

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  • $\begingroup$ All users, on February I decided to remove one of the conjectures to improve the post (more concise and focused to ask a good question for the site MathOverflow), since I was asking about what work can be done about the conjecture I accept the suitable answer, continuining my interest about this. Many thanks for all users. $\endgroup$
    – user142929
    Commented Apr 18, 2020 at 18:26

1 Answer 1

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Not a complete answer, but a bit too long for a comment: Conjecture 1 is very likely to be very difficult if true. The corresponding conjecture for general primes is open. Let $p_n$ be the $n$th prime and $g_n$ be the $n$th prime gap. If one has $\sqrt{p_{n+1}}-\sqrt{p_n} <1 $ for sufficiently large primes then one would have that $g_n = O(p^{1/2})$. The best current result (as far as I'm aware) of that form we can actually prove is that $g_n = O(p^{\frac{3}{4} + \epsilon})$ for any $\epsilon>0$ (due to Nikolai Chudakov)(Edit: see GH's comment below that we have a bound that is $O(p^{\frac{21}{40}})$ due to Baker-Harman-Pintz). Your conjecture would imply that $g_n = O(p^{1/2})$ and is seems to be substantially stronger. Since we have that $R_n$ is in general very close to $p_{2n}$ your conjecture isn't implausible, but likely to be stronger than what we can currently prove.

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  • $\begingroup$ Many thanks for your nice answer. I'am going to study it, and as soon I can I'm going to upvote it. If you want feel free to add feeback about the second of the conjectures. $\endgroup$
    – user142929
    Commented Feb 4, 2020 at 18:40
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    $\begingroup$ Chudakov's bound was improved several times. Currently we know that $g_n=O(p_n^{21/40})$. This was proved by Baker-Harman-Pintz (2000). $\endgroup$
    – GH from MO
    Commented Feb 6, 2020 at 8:08
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    $\begingroup$ @user142929: I have no time for MO now. At any rate, I suggest that you ask this as a separate question. $\endgroup$
    – GH from MO
    Commented Feb 9, 2020 at 15:32
  • $\begingroup$ Possibly unrelated to my question, is I think that a similar Firoozbakht's conjecture for Ramanujan primes maybe is feasible. I don't know if this proposal that I evoke have the best mathematical content and maybe it is in the literature, since I know that Zhi-Wei Sun studied in articles many similar inequalities involving arithmetic functions (I know his preprint in arXiv arXiv:1208.2683 of one of his papers, and I don't know if he, or other mathematician, studied the inequality that I evoke). Isn't required a response, just I add it if you want to think about it. $\endgroup$
    – user142929
    Commented Mar 13, 2020 at 21:02
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    $\begingroup$ In past week I've removed the mentioned comment (as outdated). I'm sorry for my insistence in previous weeks and many thanks for your patience @GHfromMO $\endgroup$
    – user142929
    Commented Apr 25, 2020 at 14:55

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