Open Access
Bounded gaps between primes
Andrew Granville,Yiliang Zhang +1 more
- 01 Jan 2013
TL;DR: 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.
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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 co-operative team, polymath8, collaborating only on-line, 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.
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Citations
Variants of the Selberg sieve, and bounded intervals containing many primes
TL;DR: In particular, this paper showed that for any admissible triple (h1,h2,h3), there are infinitely many n for which at least two of n+h 1,n+h 2,h 3 are prime, and also showed that either the twin prime conjecture holds or the even Goldbach conjecture is asymptotically true if one allows an additive error of at most 2, or both.
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The "bounded gaps between primes" Polymath project - a retrospective
TL;DR: In a recent breakthrough paper of Zhang, a finite upper bound was obtained for the first time on $H_1$; more specifically, Zhang showed that $H1 \leq 70000000$ as mentioned in this paper.
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TL;DR: This tool permits the user to visualize the steps of the RSA cipher, do encryption and decryption, learn simple factorization algorithms, and perform some elementary attacks.
On massive sets for subordinated random walks
TL;DR: In this paper, the authors studied the massive (reccurent) sets with respect to a simple random walk on the integer lattice and showed that some proper subsets of the set of primes are massive whereas others are non-massive.
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A Study of Relationship Among Goldbach Conjecture, Twin Prime and Fibonacci Number
TL;DR: The author will extend to introduce the relationsip among Goldbach conjecture, twin prime and Fibonacci number to completely lists all combinations of twin prime in Goldach conjecture.
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