In this post we denote for integers $n\geq 1$ the $n$-th Ramanujan prime as $R\_n$ (thus the sequence [*A104272*](https://oeis.org/search?q=A104272&language=english&go=Search) 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.*](https://en.wikipedia.org/wiki/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.



A conjecture about an inequality that involve Ramanujan primes
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. 


