%0 Generic %1 sondow2009ramanujan %A Sondow, Jonathan %D 2009 %K postulate ramanujan mathematics %T Ramanujan Primes and Bertrand's Postulate %U http://arxiv.org/abs/0907.5232 %X The $n$th Ramanujan prime is the smallest positive integer $R_n$ such that if $x R_n$, then there are at least $n$ primes in the interval $(x/2,x$. For example, Bertrand's postulate is $R_1 = 2$. Ramanujan proved that $R_n$ exists and gave the first five values as 2, 11, 17, 29, 41. In this note, we use inequalities of Rosser and Schoenfeld to prove that $2n 2n < R_n < 4n łog 4n$ for all $n$, and we use the Prime Number Theorem to show that $R_n$ is asymptotic to the $2n$th prime. We also estimate the length of the longest string of consecutive Ramanujan primes among the first $n$ primes, explain why there are more twin Ramanujan primes than expected, and make three conjectures (the first has since been proved by S. Laishram).

@misc{sondow2009ramanujan, abstract = {The $n$th Ramanujan prime is the smallest positive integer $R_n$ such that if $x \ge R_n$, then there are at least $n$ primes in the interval $(x/2,x]$. For example, Bertrand's postulate is $R_1 = 2$. Ramanujan proved that $R_n$ exists and gave the first five values as 2, 11, 17, 29, 41. In this note, we use inequalities of Rosser and Schoenfeld to prove that $2n \log 2n < R_n < 4n \log 4n$ for all $n$, and we use the Prime Number Theorem to show that $R_n$ is asymptotic to the $2n$th prime. We also estimate the length of the longest string of consecutive Ramanujan primes among the first $n$ primes, explain why there are more twin Ramanujan primes than expected, and make three conjectures (the first has since been proved by S. Laishram).}, added-at = {2013-12-23T07:00:51.000+0100}, author = {Sondow, Jonathan}, biburl = {https://www.bibsonomy.org/bibtex/27ca0b8e1d79126cac0435dcdd6deb629/aeu_research}, description = {Ramanujan Primes and Bertrand's Postulate}, interhash = {5ca5409f3a0c85e59e8382958199a116}, intrahash = {7ca0b8e1d79126cac0435dcdd6deb629}, keywords = {postulate ramanujan mathematics}, note = {cite arxiv:0907.5232Comment: 7 pages, cited Shapiro's book for Ramanujan's proof of Bertrand's Postulate}, timestamp = {2013-12-24T01:10:11.000+0100}, title = {Ramanujan Primes and Bertrand's Postulate}, url = {http://arxiv.org/abs/0907.5232}, year = 2009 }