# Ramanujan Primes and Bertrand's Postulate

@article{Sondow2009RamanujanPA,
title={Ramanujan Primes and Bertrand's Postulate},
author={Jonathan Sondow},
journal={The American Mathematical Monthly},
year={2009},
volume={116},
pages={630 - 635}
}
• J. Sondow
• Published 29 July 2009
• Mathematics
• The American Mathematical Monthly
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…
• Mathematics
• 2011
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 $R_n < p_{3n}$
The $n$th Ramanujan prime is the smallest positive integer $R_n$ such that for all $x \geq R_n$ the interval $(x/2, x]$ contains at least $n$ primes. In this paper we undertake a study of the
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• 2016
For $$n\ge 1$$n≥1, the nth Ramanujan prime is defined as the least positive integer $$R_{n}$$Rn such that for all $$x\ge R_{n}$$x≥Rn, the interval $$(\frac{x}{2}, x]$$(x2,x] has at least n primes.
In this paper, we establish several results concerning the generalized Ramanujan primes. For $$n\in \mathbb {N}$$n∈N and $$k \in \mathbb {R}_{> 1}$$k∈R>1, we give estimates for the $$n$$nth
• Mathematics
Glasnik Matematicki
• 2018
For $n\ge 1$, the $n^{\rm th}$ Ramanujan prime is defined as the smallest positive integer $R_n$ such that for all $x\ge R_n$, the interval $(\frac{x}{2}, x]$ has at least $n$ primes. We show that
• Mathematics
• 2014
In 1845, Bertrand conjectured that for all integers x ≥ 2, there exists at least one prime in (x∕2, x]. This was proved by Chebyshev in 1860 and then generalized by Ramanujan in 1919. He showed that
For n ≥ 1, the nth Ramanujan prime is defined to be the smallest positive integer Rn with the property that if x ≥ Rn, then $\pi(x)-\pi(\frac{x}{2})\ge n$ where π(ν) is the number of primes not
This work states that for each > 0, there exists an N such that Rn < p[2n(1+ )] for all n > N .
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• 2020
The degree of insulation $D(p)$ of a prime $p$ is defined as the largest interval around the prime $p$ in which no other prime is present. Based on this, the $n$-th prime $p_{n}$ is said to be
• Mathematics, Computer Science
ArXiv
• 2014
The explicit construction of weak $\left(n,\ell,\gamma\right)$-sharing set families can be used to obtain a parallelizable pseudorandom number generator with a low memory footprint by using the pseudorrandom number generator of Nisan and Wigderson.

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For n ≥ 1, the nth Ramanujan prime is defined to be the smallest positive integer Rn with the property that if x ≥ Rn, then $\pi(x)-\pi(\frac{x}{2})\ge n$ where π(ν) is the number of primes not
Perhaps the title "Ramanujan and the birth of Probabilistic Number Theory" would have been more appropriate and personal, but since Ramanujan's work influenced me greatly in other subjects too, I
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### The nth prime is greater than n ln n

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