Large gaps between primes
نویسندگان
چکیده
منابع مشابه
Chains of Large Gaps between Primes
Let pn denote the n-th prime, and for any k > 1 and sufficiently large X , define the quantity Gk(X) := max pn+k6X min(pn+1 − pn, . . . , pn+k − pn+k−1), which measures the occurrence of chains of k consecutive large gaps of primes. Recently, with Green and Konyagin, the authors showed that G1(X) ≫ logX log logX log log log logX log log logX for sufficiently large X . In this note, we combine t...
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Let G(x) denote the largest gap between consecutive grimes below x, In a series of papers from 1935 to 1963, Erdos, Rankin, and Schonhage showed that G(x) :::: (c + o( I)) logx loglogx log log log 10gx(loglog logx) -2, where c = eY and y is Euler's constant. Here, this result is shown with c = coeY where Co = 1.31256... is the solution of the equation 4/ Co e -4/co = 3 . The principal new tool ...
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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 ...
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It is proved that lim inf n→∞ (pn+1 − pn) < 7× 10, where pn is the n-th prime. Our method is a refinement of the recent work of Goldston, Pintz and Yildirim on the small gaps between consecutive primes. A major ingredient of the proof is a stronger version of the Bombieri-Vinogradov theorem that is applicable when the moduli are free from large prime divisors only (see Theorem 2 below), but it ...
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Let pn denotes the n-th prime. We prove that max p n+16X (pn+1 − pn) ≫ logX log logX log log log logX log log logX for sufficiently large X , improving upon recent bounds of the first three and fifth authors and of the fourth author. Our main new ingredient is a generalization of a hypergraph covering theorem of Pippenger and Spencer, proven using the Rödl nibble method. CONTENTS
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ژورنال
عنوان ژورنال: Annals of Mathematics
سال: 2016
ISSN: 0003-486X
DOI: 10.4007/annals.2016.183.3.3