Long gaps between primes

Abstract

Let pn denote the n-th prime. We prove that max p n+1 X (pn+1 − pn) ≫ 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.

The ternary Goldbach problem with a prime and two isolated primes

“…For a long period of time the improvements of the result (1.2) only involved the constant C ( [12], [15], [18], [20]). A famous prize problem of Paul Erdős was the improvement of the order of magnitude of the function g. This was achieved only recently ( [6], [7], [13]). The latest result ( [7]) is:…”

“…A famous prize problem of Paul Erdős was the improvement of the order of magnitude of the function g. This was achieved only recently ( [6], [7], [13]). The latest result ( [7]) is:…”

“…Our construction is mainly based on the ideas of the paper [7]. Also our definitions are mainly taken from [7].…”

“…Also our definitions are mainly taken from [7]. Let c 1 , c 2 be fixed positive constants to be specified later.…”

Geometric progression-free sequences with small gaps

“…Most recently, Ford, Green, Konyagin, Maynard, and Tao combined the previous methods and showed p i+1 − p i ≫ log p i log log p i log log log log p i log log log p i for infinitely many i and an effective implied constant, removing a log log log p i factor from the denominator. For a history of these results and the current progress, see their paper [7].…”