Primes in tuples I

  title={Primes in tuples I},
  author={D. A. Goldston and Janos Pintz and C. Y. Yildirim},
  journal={Annals of Mathematics},
We introduce a method for showing that there exist prime numbers which are very close together. The method depends on the level of distribution of primes in arithmetic progressions. Assuming the Elliott-Halberstam conjecture, we prove that there are infinitely often primes differing by 16 or less. Even a much weaker conjecture implies that there are infinitely often primes a bounded distance apart. Unconditionally, we prove that there exist consecutive primes which are closer than any… 
Small Gaps Between Almost Primes, the Parity Problem, and Some Conjectures of Erdős on Consecutive Integers
In a previous paper, the authors proved that in any system of three linear forms satisfying obvious necessary local conditions, there are at least two forms that infinitely often assume E 2 -values;
Are there arbitrarily long arithmetic progressions in the sequence of twin primes? II
In an earlier work it was shown that the Elliott-Halberstam conjecture implies the existence of infinitely many gaps of size at most 16 between consecutive primes. In the present work we show that
Primes in tuples IV: Density of small gaps between consecutive primes
α e−t dt as x→∞, for any two fixed real numbers β > α ≥ 0. Gallagher’s calculation [Ga] shows that this conjecture can be deduced from the Hardy–Littlewood prime k-tuples conjecture (see [S2]). Hence
Patterns of Primes in Arithmetic Progressions
After the proof of Zhang about the existence of infinitely many bounded gaps between consecutive primes the author showed the existence of a bounded d such that there are arbitrarily long arithmetic
Abstract. We use Maynard’s methods to show that there are bounded gaps between primes in the sequence {⌊nα⌋}, where α is an irrational number of finite type. In addition, given a superlinear function
Strings of congruent primes in short intervals II
Abstract. Let p1 = 2, p2 = 3, . . . be the sequence of all primes. Let ǫ be an arbitrarily small but fixed positive number, and fix a coprime pair of integers q > 3 and a. We will establish a lower
The existence of small prime gaps in subsets of the integers
We consider the problem of finding small prime gaps in various sets . Following the work of Goldston–Pintz–Yildirim, we will consider collections of natural numbers that are well-controlled in
Small gaps between primes
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
Small gaps between products of two primes
Let qn denote the nth number that is a product of exactly two distinct primes. We prove that qn+1 − qn ⩽ 6 infinitely often. This sharpens an earlier result of the authors, which had 26 in place of
Recently, Yitang Zhang proved the existence of a finite bound B such that there are infinitely many pairs pn, pn 1 of consecutive primes for which pn 1 pn ¤ B. This can be seen as a massive


Let p n denote the n th prime. Goldston, Pintz, and Yildirim recently proved that li m in f (pn+1 ― p n ) n→∞ log p n = 0. We give an alternative proof of this result. We also prove some
The difference of consecutive primes
is greater than (c2/2)n. A simple calculation now shows that the primes satisfying (4) also satisfy the first inequality of (3) i΀ = e(ci) is chosen small enough. The second inequality of (3) is
One of the formulations of the prime number theorem is the statement that the number of primes in an interval (n, n + ft], averaged over n ^ JV, tends to the limit A, when JV and h tend to infinity
Small gaps between primes exist
In the recent preprint (3), Goldston, Pintz, and Yoldorom established, among other things, (0) liminf n→∞ pn+1 − pn log pn = 0, with pn the nth prime. In the present article, which is essentially
Higher correlations of divisor sums related to primes III: small gaps between primes
We use divisor sums to approximate prime tuples and moments for primes in short intervals. By connecting these results to classical moment problems we are able to prove that, for any η > 0, a
A heuristic asymptotic formula concerning the distribution of prime numbers
Suppose fi, f2, -*, fk are polynomials in one variable with all coefficients integral and leading coefficients positive, their degrees being hi, h2, **. , hk respectively. Suppose each of these
On Bombieri and Davenport's theorem concerning small gaps between primes
§1. Introduction . In this paper we give a new proof of a theorem of Bombieri and Davenport [2, Theorem 1]. Let t (– k ) = t ( k ) be real, where e ( u ) = e 2 πiu . Let p and p ' denote primes, k an
The difference between consecutive prime numbers. II
In a previous paper, under the same title, I considered the problem of how far apart two consecutive primes can be. The present paper is concerned with the opposite question. How near together can
Small gaps between prime numbers: The work of Goldston-Pintz-Yildirim
In early 2005, Dan Goldston, Janos Pintz, and Cem Yildirim [12] made a spectacular breakthrough in the study of prime numbers. Resolving a long-standing open problem, they proved that there are
Higher correlations of divisor sums related to primes II: variations of the error term in the prime number theorem
We calculate the triple correlations for the truncated divisor sum λR(n). The λR(n) behave over certain averages just as the prime counting von Mangoldt function Λ(n) does or is conjectured to do. We