Firoozbakht's conjecture and maximal gaps

Question

In the Wikipedia article, it seems to me as if it's implied that it is enough to check the conjecture only for maximal gaps (numbers $n$ s.t. $\forall k<n:g_n>g_k$).

I.e it holds that $P_{k+1}<P_k^{1+\frac{1}{k}}\Rightarrow P_{m+1}<P_m^{1+\frac{1}{m}}$ where $P_{k+1}-P_{k}$ is a maximal gap and $m<k$. (Or some similar statement that involves maximal gaps.)

Is this really the case or perhaps I misread the Wikipedia article? I tried to prove it using high school manipulations, but without much success.

Answer 1

Define $q(x)$ to be the smallest prime greater than $x$ and $$ f(p)=\begin{cases}1&\text{, }p\text{ is prime and }q(p)<p^{1+1/\pi(p)}\\ 0&\text{, otherwise}\end{cases} $$

Then Firoozbakht's conjecture is that $f(p)=1$ for all primes $p$. You seem to be asking if $f(p)=1$ for all primes $p$ beginning a maximal gap implies Firoozbakht's conjecture, and it does.

I should also mention that Firoozbakht's conjecture is believed to be false. It is expected that for any $k<2/e^\gamma=1.1229\ldots$ there are infinitely many prime gaps of length greater than $kp\log^2p$ and Firoozbakht's conjecture fails if there is even one with $k\ge1.$ It was a reasonable conjecture in 1982 when it was first posed but advances in the mid-80s and later have convinced number theorists that it's highly unlikely.

Answer 2

The implication "if a certain sufficient condition $(*)$ for Firoozbakht's conjecture is true for a maximal gap starting at a prime $p_k$, $k>9$, then it is also true for all prime gaps between this maximal gap and the next one" is shown in arXiv:1506.03042 (Journal of Integer Sequences, 18, 2015, article 15.11.2) Upper bounds for prime gaps related to Firoozbakht's conjecture; see Section 4, Theorem 3 (page 4) and Remark (i) (page 5). Namely, the implication is based on the following sufficient condition for Firoozbakht's conjecture (Theorem 3): if $$p_{k+1}-p_k<\log^2 p_k - \log p_k - 1.17 \quad (*)$$ for $k>9$, then $p_{k+1}<p_k^{1+1/k}$. Just observe that $\log^2 p_k - \log p_k - 1.17$ is an increasing function of $p_k$ and use the definition of maximal gaps.