Ramanujan Primes: Bounds, Runs, Twins, and Gaps

Details Ramanujan Primes: Bounds, Runs, Twins, and Gaps Ramanujan Primes: Bounds, Runs, Twins, and Gaps

2011-05-11

arxiv.org/abs/1105.2249v2

J. Integer Seq. 14 (2011) Article 11.6.2

Mathematics

Number Theory

11 pages, 3 tables. Corrected mistakes in the published version of Table 1, added corresponding Acknowledgment

Scientific paper

The $n$th Ramanujan prime is the smallest positive integer $R_n$ such that if $x \ge R_n$, then the interval $(x/2,x]$ contains at least $n$ primes. We sharpen Laishram's theorem that $R_n < p_{3n}$ by proving that the maximum of $R_n/p_{3n}$ is $R_5/p_{15} = 41/47$. We give statistics on the length of the longest run of Ramanujan primes among all primes $p<10^n$, for $n\le9$. We prove that if an upper twin prime is Ramanujan, then so is the lower; a table gives the number of twin primes below $10^n$ of three types. Finally, we relate runs of Ramanujan primes to prime gaps. Along the way we state several conjectures and open problems. The Appendix explains Noe's fast algorithm for computing $R_1,R_2,...,R_n$.

Affiliated with

Also associated with

No associations

Rating

Ramanujan Primes: Bounds, Runs, Twins, and Gaps does not yet have a rating. At this time, there are no reviews or comments for this scientific paper.

If you have personal experience with Ramanujan Primes: Bounds, Runs, Twins, and Gaps, we encourage you to share that experience with our LandOfFree.com community. Your opinion is very important and Ramanujan Primes: Bounds, Runs, Twins, and Gaps will most certainly appreciate the feedback.

Rate now

Comments { 0 }

Profile ID: LFWR-SCP-O-612032