Primes in tuples II
نویسندگان
چکیده
منابع مشابه
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 ap...
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In this paper, we show that if p is a prime and ifA = {a1, a2, . . . , am} is a set of positive integers with the property that aiaj +p is a perfect square for all 1 ≤ i < j ≤ m, then m < 3 · 2168. More generally, when p is replaced by a squarefree integer n, the inequality m ≤ f(ω(n)) holds with some function f , where ω(n) is the number of prime divisors of n. We also give upper bounds for m ...
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متن کاملSmall Gaps between Primes Ii (preliminary)
We examine an idea for approximating prime tuples. 1. Statement of results (Preliminary) In the present work we will prove the following result. Let pn denote the nth prime. Then (1.1) lim inf n→∞ (pn+1 − pn) log pn(log log pn)−1 log log log log pn < ∞. Further we show that supposing the validity of the Bombieri–Vinogradov theorem up to Q ≤ X with any level θ > 1/2 we have bounded differences b...
متن کاملar X iv : 0 71 0 . 27 28 v 1 [ m at h . N T ] 1 5 O ct 2 00 7 PRIMES IN TUPLES II
We prove that lim inf n→∞ pn+1 − pn √ log pn(log log pn) < ∞, where pn denotes the n prime. Since on average pn+1 −pn is asymptotically log pn, this shows that we can always find pairs of primes much closer together than the average. We actually prove a more general result concerning the set of values taken on by the differences p− p between primes which includes the small gap result above.
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ژورنال
عنوان ژورنال: Acta Mathematica
سال: 2010
ISSN: 0001-5962
DOI: 10.1007/s11511-010-0044-9