Long gaps between primes
Abstract
Let $p_n$ denotes the $n$th prime. We prove that $$\max_{p_{n+1} \leq X} (p_{n+1}p_n) \gg \frac{\log X \log \log X\log\log\log\log X}{\log \log \log X}$$ for sufficiently large $X$, improving upon recent bounds of the first three and fifth authors and of the fourth author. Our main new ingredient is a generalization of a hypergraph covering theorem of Pippenger and Spencer, proven using the Rödl nibble method.
 Publication:

arXiv eprints
 Pub Date:
 December 2014
 DOI:
 10.48550/arXiv.1412.5029
 arXiv:
 arXiv:1412.5029
 Bibcode:
 2014arXiv1412.5029F
 Keywords:

 Mathematics  Number Theory;
 Mathematics  Combinatorics;
 11N05;
 11N35;
 05C70
 EPrint:
 (i) in the introduction, we added a corollary about the least prime in an arithmetic progression