Erdős $5000 Problem on Prime Gaps

Definitions and mathematical statement

Shared notation.

$\mathbb N=\{1,2,\ldots\}$, and $\mathbb Z,\mathbb Q,\mathbb R,\mathbb C$ denote the integers, rationals, reals and complex numbers. Any other conventions are specified locally.

Let $p_n$ be the $n$-th prime and $d_n=p_{n+1}-p_n$. Let $\log$ denote the natural logarithm, and define its iterates $\log_1x=\log x$ and $\log_{j+1}x=\log(\log_jx)$, whenever these expressions exist. For sufficiently large $x$, put \[ R(x)=\frac{\log x\,\log_2x\,\log_4x} {(\log_3x)^2}>0. \] This is the Rankin scale in the recorded question. Ford, Green, Konyagin and Tao proved the arbitrary-constant growth result in the cited theorem, settling this historical target.

Statement recorded in the supplied source.

Is it true that, for every real constant $c>0$ and every integer $N\ge1$, there is an integer $n\ge N$, large enough that $R(n)$ is defined and positive, satisfying \[ d_n>c\,R(n)? \] Equivalently, is $\limsup_{n\to\infty}d_n/R(n)=\infty$? [S2]

Short English statement

Do consecutive prime gaps exceed every fixed multiple of the displayed four-logarithm Rankin scale infinitely often? This historical question was answered affirmatively.

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 1861 (GUY-A8a). CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 Kevin Ford, Ben Green, Sergei Konyagin and Terence Tao, Large gaps between consecutive prime numbers, Annals of Mathematics 183 (2016), 935--974; arXiv:1408.4505v2, Section 1, Theorem 1 and the arbitrary-constant Rankin-scale result. https://arxiv.org/abs/1408.4505.