# Is this conjecture on the $n$th record prime gap “true by accident”?

Let $R(n)$ be the $n$th record gap between primes; $R(n)=\mbox{A005250}(n)$ in OEIS.

The paper arXiv:1709.05508 conjectures, among other things, that $$R(n) = O(n^2) \tag{1}$$ and, more specifically, $$R(n) \le n^2. \tag{2}$$

(The heuristic reasoning in arXiv:1709.05508 is based on (a) Cramer's conjecture and (b) a conjecture that record prime gaps occur more often than records in an i.i.d. random sequence of comparable length.) From computations, we know that the first 77 record (maximal) prime gaps (between primes below $10^{19}$) do satisfy $(2)$.

Can we reasonably say that $(1)$ and $(2)$ are true by accident (and therefore not amenable to proof)?

Edit: Many thanks @Gerhard and @reuns for your insights! Based on more thinking, I am now sure that in Cramer's probabilistic model of primes we can prove: with probability 1, the true order of magnitude of the $n$th record prime gap is $O(n^2)$ (I haven't found a reference yet, though). So, Gerhard's assessment below would mean that the true order of $n$th record gap between ordinary primes is less than that for Cramer's probabilistic primes.

• Thank you Gerhard. I do agree, we may just know too little, and statements (1) and (2) may be weak approximations to the truth. However, if Cramer's conjecture is true and if another conjecture in arXiv:1709.05508 is also true, namely, if the number of record gaps observed between primes below $x$ is $O(\log x)$, then $O(n^2)$ in (1) might well be the true order of $R(n)$. – Alex Jan 5 '18 at 6:13
• I compared the truth with the random model for the primes ($X_n = 1_{n \text{ is prime}}$ are independent random variables with $P[X_n = 1] = \frac{1}{\log n}$) and the difference doesn't look too high. @Alex – reuns Jan 5 '18 at 13:34