Number of solutions of some diofantine inequalities in special primes

Author(s):  A.P. Naumenko, Lomonosov Moscow State University, Moscow, Russia, naumenko.anton90@gmail.com

Issue:  Volume 50, № 3

Rubric:  Mathematics

Annotation:  The question of whether there exist infinitely many twin primes has been one of the great open questions in number theory for many years. This is the content of the twin prime conjecture (non proof), which states that there are infinitely many primes p such that p+2 is also prime. In their celebrated paper Goldston, Pintz and Yildirim introduced a new method for counting tuples of primes, and this allowed them to show that lim┬n⁡〖(p_(n+1)-p_n)/log⁡〖p_n 〗 〗=0. In 2011 the recent breakthrough of Zhang managed to extend this work to prove lim┬n⁡〖(p_(n+1)-p_n )≤7∙〖10〗^7 〗thereby establishing for the first time the existence of infinitely many bounded gaps between primes. Later James Maynard has succeeded in reducing the Zhang’s bound to 600. The recent polymath project has succeeded in reducing the bound to 246, by optimizing Zhang's arguments and introducing several new refinements. Let P_0 is the set of primes p satisfies 0≤{√p}≤0.5. We introduce a refinement of the GPY sieve method for studying small gaps between «special» primes. This refinement avoids previous limitations of the method, and allows us to show that lim┬n⁡〖(p_(n+1)-p_n )≤478830〗

Keywords:  gaps between primes, Selberg sieve

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