منابع مشابه
Small Gaps between Primes or Almost Primes
Let pn denote the nth prime. Goldston, Pintz, and Yıldırım recently proved that lim inf n→∞ (pn+1 − pn) log pn = 0. We give an alternative proof of this result. We also prove some corresponding results for numbers with two prime factors. Let qn denote the nth number that is a product of exactly two distinct primes. We prove that lim inf n→∞ (qn+1 − qn) ≤ 26. If an appropriate generalization of ...
متن کاملSmall gaps between primes
The twin prime conjecture states that there are infinitely many pairs of distinct primes which differ by 2. Until recently this conjecture had seemed to be out of reach with current techniques. However, in 2013, the author proved that there are infinitely many pairs of distinct primes which differ by no more than B with B = 7 · 107. The value of B has been considerably improved by Polymath8 (a ...
متن کاملSmall Gaps between Primes I
Finding mathematical proofs for easily observed properties of the distribution of prime numbers is a difficult and often humbling task, at least for the authors of this paper. The twin prime conjecture is a famous example of this, but we are concerned here with the much more modest problem of proving that there are arbitrarily large primes that are “unusually close ” together. Statistically thi...
متن کاملSmall Gaps between Primes Exist
In the preprint [3], Goldston, Pintz, and Yıldırım established, among other things, (0) lim inf n→∞ pn+1 − pn log pn = 0, with pn the nth prime. In the present article, which is essentially self-contained, we shall develop a simplified account of the method used in [3]. We include a short expository last section. Key word: Prime number. 1. Basic lemma. In this section we shall prove an asymptot...
متن کاملSmall 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 6. More generally, we prove that if ν is any positive integer, then (qn+ν − qn) νeν−γ(1 + o(1)) infinitely often. We also prove several other related results on the representation of numbers with exact...
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ژورنال
عنوان ژورنال: Transactions of the American Mathematical Society
سال: 2009
ISSN: 0002-9947
DOI: 10.1090/s0002-9947-09-04788-6