Long gaps between primes

Abstract

Let $p_n$ denotes the $n$-th prime. We prove that $$\underset{p_{n+1}\le X}{\max }(p_{n+1}-p_n)\gg \frac{\log X\log \log X\log \log \log \log X}{\log \log \log X}$$ 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.

(i) in the introduction, we added a corollary about the least prime in an arithmetic progression; (ii) relaxed the hypotheses of Cor. 4: now sets P' and Q' may be arbitrary sets, not necessarily sets of primes; this has an application (arXiv:1607.02543); (iii) updated many references

Cited by in corpus (4)

• Large gaps between consecutive prime numbers
• Bounded gaps between primes in short intervals
• Geometric Progression-Free Sequences with Small Gaps
• A lower bound for the least prime in an arithmetic progression