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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 ElliottHalberstam conjecture, we prove that there are infinitely often primes differing by 16 or less. E ..."
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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 ElliottHalberstam 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, pn+1 − pn lim inf =0. n→ ∞ log pn We will quantify this result further in a later paper (see (1.9) below).
Small gaps between prime numbers: the work of GoldstonPintzYıldırım
, 2000
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BOUNDED GAPS BETWEEN PRODUCTS OF PRIMES WITH APPLICATIONS TO IDEAL CLASS GROUPS AND ELLIPTIC CURVES
"... and Y ld r m use a variant of the Selberg sieve to prove the existence of small gaps between E2 numbers, that is, squarefree numbers with exactly two prime factors. We apply their techniques to prove similar bounds for Er numbers for any r ≥ 2, where these numbers are required to have all of their ..."
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and Y ld r m use a variant of the Selberg sieve to prove the existence of small gaps between E2 numbers, that is, squarefree numbers with exactly two prime factors. We apply their techniques to prove similar bounds for Er numbers for any r ≥ 2, where these numbers are required to have all of their prime factors in a set of primes P. Our result holds for any P of positive density that satis es a SiegelWal sz condition regarding distribution in arithmetic progressions. We also prove a stronger result in the case that P satis es a BombieriVinogradov condition. We were motivated to prove these generalizations because of recent results of Ono [22] and Soundararajan [25]. These generalizations yield applications to divisibility of class numbers, nonvanishing of critical values of Lfunctions, and triviality of ranks of elliptic curves. 1.
Small gaps between almost primes, the parity problem and some conjectures of Erdős on consecutive integers
, 2008
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"... Using Jiang function we prove Jiang prime ktuple theorem. We prove that the HardyLittlewood prime ktuple conjecture is false. Jiang prime ktuple theorem can replace the HardyLittlewood prime ktuple conjecture. ..."
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Using Jiang function we prove Jiang prime ktuple theorem. We prove that the HardyLittlewood prime ktuple conjecture is false. Jiang prime ktuple theorem can replace the HardyLittlewood prime ktuple conjecture.
J ( ω) = Π ( P−1 − χ ( P)), (2) 2
"... Using Jiang function we prove Jiang prime ktuple theorem. We prove that the HardyLittlewood prime ktuple conjecture is false. Jiang prime ktuple theorem can replace the HardyLittlewood prime ktuple conjecture. ..."
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Using Jiang function we prove Jiang prime ktuple theorem. We prove that the HardyLittlewood prime ktuple conjecture is false. Jiang prime ktuple theorem can replace the HardyLittlewood prime ktuple conjecture.
BOUNDED GAPS BETWEEN PRIMES
"... Abstract. Recently, Yitang Zhang proved the existence of a finite bound B such that there are infinitely many pairs pn, pn 1 of consecutive primes for which pn 1 pn ¤ B. This can be seen as a massive breakthrough on the subject of twin primes and other delicate questions about prime numbers that had ..."
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Abstract. Recently, Yitang Zhang proved the existence of a finite bound B such that there are infinitely many pairs pn, pn 1 of consecutive primes for which pn 1 pn ¤ B. This can be seen as a massive breakthrough on the subject of twin primes and other delicate questions about prime numbers that had previously seemed intractable. In this article we will discuss Zhang’s extraordinary work, putting it in its context in analytic number theory, and sketch a proof of his theorem. Zhang even proved the result with B 70 000 000. A cooperative team, polymath8, collaborating only online, has been able to lower the value of B to 4680, and it seems plausible that these techniques can be pushed somewhat further, though the limit of these methods seem, for now, to be B 12. Contents
GAPS BETWEEN PRIME NUMBERS AND PRIMES IN ARITHMETIC PROGRESSIONS
, 2014
"... ... utinam intelligere possim rationacinationes pulcherrimas quae e propositione concisa ∑xiyi  6 ‖x ‖ ‖y ‖ fluunt... (after H. Cartan) 1. ..."
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... utinam intelligere possim rationacinationes pulcherrimas quae e propositione concisa ∑xiyi  6 ‖x ‖ ‖y ‖ fluunt... (after H. Cartan) 1.
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"... Using Jiang function we prove Jiang prime ktuple theorem.We find true singular series. Using the examples we prove the HardyLittlewood prime ktuple conjecture with wrong singular series.. Jiang prime ktuple theorem will replace the HardyLittlewood prime ktuple conjecture. (A) Jiang prime ktup ..."
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Using Jiang function we prove Jiang prime ktuple theorem.We find true singular series. Using the examples we prove the HardyLittlewood prime ktuple conjecture with wrong singular series.. Jiang prime ktuple theorem will replace the HardyLittlewood prime ktuple conjecture. (A) Jiang prime ktuple theorem with true singular series[1, 2]. We define the prime ktuple equation p, p � ni, (1) where 2 n, i �1, � k�1. i we have Jiang function [1, 2]
doi:10.1093/imrn/rnq124 Small Gaps Between Almost Primes, the Parity Problem, and Some Conjectures of Erdős on Consecutive Integers
, 2009
"... In a previous paper, the authors proved that in any system of three linear forms satisfying obvious necessary local conditions, there are at least two forms that infinitely often assume E2values; that is, values that are products of exactly two primes. We use this result to prove that there are inf ..."
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In a previous paper, the authors proved that in any system of three linear forms satisfying obvious necessary local conditions, there are at least two forms that infinitely often assume E2values; that is, values that are products of exactly two primes. We use this result to prove that there are infinitely many positive integers x such that both x and x+ 1 have prime factorizations of the form p21 p2p3p4. Consequently, there are infinitely many integers x that simultaneously sat