It is proved that 
lim inf n?8 (p n+1 -p n )<7×10 7 , where p n is the n -th prime. 
Our method is a refinement of the recent work of Goldston, Pintz and Yildirim on the small gaps between consecutive primes. A major ingredient of the proof is a stronger version of the Bombieri-Vinogradov theorem that is applicable when the moduli are free from large prime divisors only, but it is adequate for our purpose

/pdf/bounded-gaps-between-primes-2y8s6yn152.pdf

Bounded gaps between primes

The degree/diameter problem is to determine the largest graphs or digraphs of given maximum degree and given diameter. General upper bounds - called Moore bounds - for the order of such graphs and digraphs are attainable only for certain special graphs and digraphs. Finding better (tighter) upper bounds for the maximum possible number of vertices, given the other two parameters, and thus attacking the degree/diameter problem 'from above', remains a largely unexplored area. Constructions producing large graphs and digraphs of given degree and diameter represent a way of attacking the degree/diameter problem 'from below'. This survey aims to give an overview of the current state-of-the-art of the degree/diameter problem. We focus mainly on the above two streams of research. However, we could not resist mentioning also results on various related problems. These include considering Moore-like bounds for special types of graphs and digraphs, such as vertex-transitive, Cayley, planar, bipartite, and many others, on the one hand, and related properties such as connectivity, regularity, and surface embeddability, on the other hand.

Moore Graphs and Beyond: A survey of the Degree/Diameter Problem

On the order of magnitude of the difference between consecutive prime numbers

For any m≥1, let H
 m
 denote the quantity . A celebrated recent result of Zhang showed the finiteness of H1, with the explicit bound H1≤70,000,000. This was then improved by us (the Polymath8 project) to H1≤4680, and then by Maynard to H1≤600, who also established for the first time a finiteness result for H
 m
 for m≥2, and specifically that H
 m
 ≪m3e4m. If one also assumes the Elliott-Halberstam conjecture, Maynard obtained the bound H1≤12, improving upon the previous bound H1≤16 of Goldston, Pintz, and Yildirim, as well as the bound H
 m
 ≪m3e2m. In this paper, we extend the methods of Maynard by generalizing the Selberg sieve further and by performing more extensive numerical calculations. As a consequence, we can obtain the bound H1≤246 unconditionally and H1≤6 under the assumption of the generalized Elliott-Halberstam conjecture. Indeed, under the latter conjecture, we show the stronger statement that for any admissible triple (h1,h2,h3), there are infinitely many n for which at least two of n+h1,n+h2,n+h3 are prime, and also obtain a related disjunction asserting that either the twin prime conjecture holds or the even Goldbach conjecture is asymptotically true if one allows an additive error of at most 2, or both. We also modify the ‘parity problem’ argument of Selberg to show that the H1≤6 bound is the best possible that one can obtain from purely sieve-theoretic considerations. For larger m, we use the distributional results obtained previously by our project to obtain the unconditional asymptotic bound or H
 m
 ≪m e2m under the assumption of the Elliott-Halberstam conjecture. We also obtain explicit upper bounds for H
 m
 when m=2,3,4,5.

/pdf/variants-of-the-selberg-sieve-and-bounded-intervals-49ir8udy7e.pdf

Variants of the Selberg sieve, and bounded intervals containing many primes

Let A be a thin set in Z n , and A N the intersection of A with the ball of diameter N centred at the origin. When n = 1 we have seen in §9.7 (as a consequence of Siegel’s theorem) that |A N | = O(N 1/2) when N → ∞. To prove a similar result when n ≥ 2 one needs a different method, based on the large sieve inequality (cf. [Co]).

The Large Sieve

We introduce a renement of the GPY sieve method for studying prime k-tuples and small gaps between primes. This renement avoids previous limitations of the method and allows us to show that for each k, the prime k-tuples conjecture holds for a positive proportion of admissible k-tuples. In particular, lim infn(pn+m pn) <1 for every integer m. We also show that lim inf(pn+1 pn) 600 and, if we assume the Elliott-Halberstam conjecture, that lim infn(pn+1 pn) 12 and lim infn(pn+2 pn) 600.

/pdf/small-gaps-between-primes-46rsksn6bx.pdf

Small gaps between primes

Update: This work reproduces an earlier result of Peck, which the author was initially unaware of. The method of the proof is essentially the same as the original work of Peck. There are no new results. 
We show that the sum of squares of differences between consecutive primes $\sum_{p_n\le x}(p_{n+1}-p_n)^2$ is bounded by $x^{5/4+{\epsilon}}$ for $x$ sufficiently large and any fixed ${\epsilon}>0$. The proof relies on utilising various mean-value estimates for Dirichlet polynomials.

On the difference between consecutive primes

We prove a generalization of the author's work to show that any subset of the primes which is `well-distributed' in arithmetic progressions contains many primes which are close together. Moreover, our bounds hold with some uniformity in the parameters. As applications, we show there are infinitely many intervals of length $(\log{x})^{\epsilon}$ containing $\gg_\epsilon \log\log{x}$ primes, and show lower bounds of the correct order of magnitude for the number of strings of $m$ congruent primes with $p_{n+m}-p_n\le \epsilon\log{x}$.

Dense clusters of primes in subsets

Let pn denote the n-th prime. We prove that max/pn+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 Rodl nibble method.

Long gaps between primes

We prove a generalization of the author’s work to show that any subset of the primes which is ‘well distributed’ in arithmetic progressions contains many primes which are close together. Moreover, our bounds hold with some uniformity in the parameters. As applications, we show there are infinitely many intervals of length containing primes, and show lower bounds of the correct order of magnitude for the number of strings of congruent primes with .

/pdf/dense-clusters-of-primes-in-subsets-4pat1dtfl4.pdf

James Maynard

Papers

Small gaps between primes

On the difference between consecutive primes

Dense clusters of primes in subsets

Long gaps between primes

Dense clusters of primes in subsets