• Corpus ID: 7077803

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

@article{Sondow2011RamanujanPB,
title={Ramanujan Primes: Bounds, Runs, Twins, and Gaps},
author={Jonathan Sondow and John W. Nicholson and Tony D. Noe},
journal={arXiv: Number Theory},
year={2011}
}
• Published 1 May 2011
• Mathematics
• arXiv: Number Theory
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…
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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.
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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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
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Sondow et al have studied Ramanujan primes (RPs) and observed numerically that, while half of all primes are RPs asymptotically, one obtains runs of consecutives RPs (resp. non-RPs) which are
The Prime Number Theorem is used to show that $R_n$ is asymptotic to the $2n$th prime and the length of the longest string of consecutive Ramanujan primes among the first $n$ primes is estimated.
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
1. How Many Prime Numbers Are There?.- I. Euclid's Proof.- II. Kummer's Proof.- III. Polya's Proof.- IV. Euler's Proof.- V. Thue's Proof.- VI. Two-and-a-Half Forgotten Proofs.- A. Perott's Proof.- B.
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THIS book must be welcomed most warmly into X the select class of Oxford books on pure mathematics which have reached a second edition. It obviously appeals to a large class of mathematical readers.
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The On-Line Encyclopedia of Integer Sequences (or OEIS) is a database of some 130000 number sequences which serves as a dictionary, to tell the user what is known about a particular sequence and is widely used.
Mark F. Schilling is Associate Professor at California State University, Northridge. He received his BA and M.A. in mathe? matics at the University of California at San Diego and his doctorate was

### dénominateurs sont " nombres premiers jumeaux " est convergente ou finie

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### Inégalités explicites pour ψ(X), θ(X), π(X) et les nombres premiers

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