Janos Pintz

Primes in tuples I

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 arbitrarily small multiple of the average spacing, that is, lim inf n→∞ Pn+1-Pn/log Pn/log = 0. We will quantify this result further in a later paper.

Extremal Uncrowded Hypergraphs

Abstract Let G be a (k + 1)-graph (a hypergraph with each hyperedge of size k + 1) with n vertices and average degreee t. Assume k ⪡ t ⪡ n. If G is uncrowded (contains no cycle of size 2, 3, or 4) then there exists and independent set of size c k ( n t )( ln t) 1 k .

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 - qnle; 6 infinitely often. This sharpens an earlier result of the authors, which had 26 in place of 6. More generally, we prove that if ? is any positive integer, then (qn+1 - qn) ≤ e-γ(1 + o(1)) infinitely often. We also prove several other related results on the representation of numbers with exactly two prime factors by linear forms.

New equidistribution estimates of Zhang type

In May 2013, Y. Zhang [52] proved the existence of infinitely many pairs of primes with bounded gaps. In particular, he showed that there exists at least one h ě 2 such that the set tp prime | p` h is primeu is infinite. (In fact, he showed this for some even h between 2 and 7ˆ 10, although the precise value of h could not be extracted from his method.) Zhang’s work started from the method of Goldston, Pintz and Yıldırım [23], who had earlier proved the bounded gap property, conditionally on distribution estimates concerning primes in arithmetic progressions to large moduli, i.e., beyond the reach of the Bombieri–Vinogradov theorem. Based on work of Fouvry and Iwaniec [11, 12, 13, 14] and Bombieri, Friedlander and Iwaniec [3, 4, 5], distribution estimates going beyond the Bombieri–Vinogradov range for arithmetic functions such as the von Mangoldt function were already known. However, they involved restrictions concerning the residue classes which were incompatible with the method of Goldston, Pintz and Yıldırım. Zhang’s resolution of this difficulty proceeded in two stages. First, he isolated a weaker distribution estimate that sufficed to obtain the bounded gap property (still

SMALL GAPS BETWEEN PRIMES OR ALMOST PRIMES

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 corresponding results for numbers with two prime factors. Let q n denote the n th number that is a product of exactly two distinct primes. We prove that lim inf(qn+1―q n ) n→∞ 26. If an appropriate generalization of the Elliott-Halberstam Conjecture is true, then the above bound can be improved to 6.

Primes in Shart Intervals.

Let π(x) stand for the number of primes not exceedingx. In the present work it is shown that if 23/42≤Θ≤1,y≤xθ andx>x(Θ) then π(x)−π(x−y)>y/(100 logx). This implies for the difference between consecutive primes the inequalitypn+1−pn≪pn23/42.

A smoothed GPY sieve

Combining the arguments developed in the works of D. A. Goldston, S. W. Graham, J. Pintz, and C. Y. Yildirim [Preprint, 2005, arXiv: math.NT/506067] and Y. Motohashi [Number theory in progress - A. Schinzel Festschrift (de Gruyter, 1999) 1053-1064] we introduce a smoothing device to the sieve procedure of Goldston, Pintz, and Yildirim (see [Proc. Japan Acad. 82A (2006) 61-65] for its simplified version). Our assertions embodied in Lemmas 3 and 4 of this article imply that a natural extension of a prime number theorem of E. Bombieri, J. B. Friedlander, and H. Iwaniec [Theorem 8 in Acta Math. 156 (1986) 203-251] should give rise infinitely often to bounded differences between primes, that is, a weaker form of the twin prime conjecture.

Polignac Numbers, Conjectures of Erdős on Gaps Between Primes, Arithmetic Progressions in Primes, and the Bounded Gap Conjecture

In the present work we prove a number of results about gaps between consecutive primes. The proofs need the method of Y. Zhang which led to the proof of infinitely many bounded gaps between primes. Several of the results refer to the so-called Polignac numbers which we define as those even integers which can be written in infinitely many ways as the difference of two consecutive primes. Others refer to several 60–70 years old conjecture of Paul Erdős about the distribution of the normalized gaps between consecutive primes and about the distribution of the ratio of consecutive primegaps. The methods involve an extended version of Zhangs method, a property of the GPY weights proved by the author a few years ago and other ideas as well.

A note on the exceptional set for Goldbach's problem in short intervals

Assume the Generalized Riemann Hypothesis and suppose thatHlog−6x→t8. Then we prove that all even integers in any interval of the form (x, x, +H) butO(H1/2log3x) exceptions are a sum of two primes.

Infinite sets of primes with fast primality tests and quick generation of large primes

Infinite sets P and Q of primes are described, P C Q. For any natural number n it can be decided if n E P in (deterministic) time O((logn)9). This answers affirmatively the question of whether there exists an infinite set of primes whose membership can be tested in polynomial time, and is a main result of the paper. Also, for every n E Q, we show how to randomly produce a proof of the primality of n. The expected time is that needed for 1 exponentiations mod n. We also show how to randomly generate k-digit integers which will be in Q with probability proportional to k-1. Combined with the fast verification of n E Q just mentioned, this gives an 0(k4) expected time algorithm to generate and certify primes in a given range and is probably the fastest method to generate large certified primes known to belong to an infinite subset. Finally, it is important that P and Q are relatively dense (at least cn2/3/ log n elements less than n). Elements of Q in a given range may be generated quickly, but it would be costly for an adversary to search Q in this range, a property that could be useful in cryptography.