Primes in tuples I
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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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...
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ژورنال
عنوان ژورنال: Annals of Mathematics
سال: 2009
ISSN: 0003-486X
DOI: 10.4007/annals.2009.170.819