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27.05.2013 · An exciting paper about gaps between prime numbers - a step closer to proving the twin prime conjecture. More links & stuff in full description below ↓↓↓Extr...

title = {Unusually large gaps between consecutive primes}, journal = {Trans. Amer. Math. Soc.}, fjournal = {Transactions of the American Mathematical Society},

A prime gap of length n is a run of n-1 consecutive composite numbers between two successive primes. Therefore, the difference between two successive primes p_k and p_(k+1) bounding a prime gap of length n is p_(k+1)-p_k=n, where p_k …

Bounded gaps between primes By Yitang Zhang Abstract It is proved that liminf n!1 (p n+1 p n) < 7 10 7; where p nis the n-th prime. Our method is a re nement of the recent work of Goldston, Pintz and Y ld r m on the small gaps between consecutive primes. A major ingredient of the proof is a stronger version of the Bombieri-Vinogradov theorem that

Bounded gaps between primes By Yitang Zhang Abstract It is proved that liminf n!1 (p n+1 p n) < 7 10 7; where p nis the n-th prime. Our method is a re nement of the recent work of Goldston, Pintz and Y ld r m on the small gaps between consecutive primes. A major ingredient of the proof is a stronger version of the Bombieri-Vinogradov theorem that

Consequence:Thus, looking at primes up to X, the average distance to the next prime is ˇlogX. So the “large” gap of size ˇlogX=loglogX was actually a small gap. Remark Most primes p X exceed X=(logX)2(say), and so logX ˇlogp. So the gap from p to the next prime is ˇlogp on average. 8 of 57 Asking the right question

From the pigeonhole principle, this implies that one has pn+1 − pn ≤ (1 + o(1))logX for some prime gap in [X,2X]. This pigeonhole bound was steadily improved ...

By the prime number theorem we know there are approximately n /log ( n) (natural log) primes less than n, so the "average gap" between primes less than n is log ( n ). But how wide of range can these gaps have? We will discuss several aspects of this question below. 2. lim inf g ( n) = 1 (?) and lim sup g ( n) = infinity

Stijn Hanson (York) Gaps Between Primes. Sieving Gaps Between Primes Beyond Bounded Gaps General Prime Constellations Distribution Conjecture (Polignac, 1843) Let k be any positive integer. Then, for in nitely many n 2N, we have that p n+1 p n = 2k. Conjecture (Dickson, 1904) Let a 1 + b 1n;a 2 + b

where pn is the n-th prime. Our method is a refinement of the recent work of Goldston, Pintz and Yıldırım on the small gaps between consecutive primes.

Our proof works by incorporating recent progress in sieve methods related to small gaps between primes into the Erdos-Rankin construction.

Even better results are possible under the Riemann hypothesis. Harald Cramér proved that the Riemann hypothesis implies the gap gn satisfies using the big O notation. (In fact this result needs only the weaker Lindelöf hypothesis, if one can tolerate an infinitesimally smaller exponent. ) Later, he conjectured that the gaps are even smaller. Roughly speaking, Cramér's conjec…

By the prime number theorem we know there are approximately n/log(n) (natural log) primes less than n, so the "average gap" between primes less than n is log(n) ...

A prime gap of length n is a run of n-1 consecutive composite numbers between two successive primes. Therefore, the difference between two successive primes ...

Large gaps between primes Pages 915-933 from Volume 183 (2016), Issue 3 by James Maynard Abstract We show that there exist pairs of consecutive primes less than x whose difference is larger than t ( 1 + o ( 1)) ( log x) ( log log x) ( log log log log x) ( log log log x) − 2 for any fixed t. This answers a well-known question of Erdős.