On gaps between numbers with a large prime factor

Large Gaps between Consecutive Prime Numbers

Let G(X) denote the size of the largest gap between consecutive primes below X . Answering a question of Erdős, we show that G(X) > f(X) logX log logX log log log logX (log log logX) , where f(X) is a function tending to infinity with X . Our proof combines existing arguments with a random construction covering a set of primes by arithmetic progressions. As such, we rely on recent work on the e...

Gaps between Prime Numbers

Let dn = Pn+i ~Pn denote the nth gap in the sequence of primes. We show that for every fixed integer A; and sufficiently large T the set of limit points of the sequence {(dn/logra, ■ • • ,dn+k-i/logn)} in the cube [0, T]k has Lebesgue measure > c(k)Tk, where c(k) is a positive constant depending only on k. This generalizes a result of Ricci and answers a question of Erdös, who had asked to prov...

Large gaps between consecutive prime numbers containing perfect powers

For any positive integer k, we show that infinitely often, perfect k-th powers appear inside very long gaps between consecutive prime numbers, that is, gaps of size ck log p log2 p log4 p (log3 p) 2 , where p is the smaller of the two primes.

Numbers with a large prime factor

Gaps between Prime Numbers and Primes in Arithmetic Progressions

The equivalence of the two formulations is clear by the pigeon-hole principle. The first one is psychologically more spectacular: it emphasizes the fact that for the first time in history, one has proved an unconditional existence result for infinitely many primes p and q constrained by a binary condition q − p = h. Remarkably, this already extraordinary result was improved in spectacular fashi...