On intervals (kn, (k + 1)n) containing a prime for all n > 1

Vladimir Shevelev; Charles R. Greathouse; Peter J.C. Moses

On intervals (kn, (k + 1)n) containing a prime for all n > 1

Vladimir Shevelev, Charles R. Greathouse, Peter J.C. Moses

Department of Mathematics

Research output: Contribution to journal › Article › peer-review

3 Scopus citations

Abstract

We study values of k for which the interval (kn, (k + 1)n) contains a prime for every n > 1. We prove that the list of such integers k includes 1, 2, 3, 5, 9, 14 and no others, at least for k ≤ 100, 000, 000. Moreover, for every known k in this list, we give a good upper bound for the smallest N_k(m), such that if n ≥ N_k(m), then the interval (kn, (k + 1)n) contains at least m primes.

Original language	English
Journal	Journal of Integer Sequences
Volume	16
Issue number	7
State	Published - 1 Jan 2013

ASJC Scopus subject areas

Discrete Mathematics and Combinatorics

Cite this

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title = "On intervals (kn, (k + 1)n) containing a prime for all n > 1",

abstract = "We study values of k for which the interval (kn, (k + 1)n) contains a prime for every n > 1. We prove that the list of such integers k includes 1, 2, 3, 5, 9, 14 and no others, at least for k ≤ 100, 000, 000. Moreover, for every known k in this list, we give a good upper bound for the smallest Nk(m), such that if n ≥ Nk(m), then the interval (kn, (k + 1)n) contains at least m primes.",

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AU - Shevelev, Vladimir

AU - Greathouse, Charles R.

AU - Moses, Peter J.C.

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Y1 - 2013/1/1

N2 - We study values of k for which the interval (kn, (k + 1)n) contains a prime for every n > 1. We prove that the list of such integers k includes 1, 2, 3, 5, 9, 14 and no others, at least for k ≤ 100, 000, 000. Moreover, for every known k in this list, we give a good upper bound for the smallest Nk(m), such that if n ≥ Nk(m), then the interval (kn, (k + 1)n) contains at least m primes.

AB - We study values of k for which the interval (kn, (k + 1)n) contains a prime for every n > 1. We prove that the list of such integers k includes 1, 2, 3, 5, 9, 14 and no others, at least for k ≤ 100, 000, 000. Moreover, for every known k in this list, we give a good upper bound for the smallest Nk(m), such that if n ≥ Nk(m), then the interval (kn, (k + 1)n) contains at least m primes.

UR - https://www.scopus.com/pages/publications/84881046447

M3 - Article

AN - SCOPUS:84881046447

SN - 1530-7638

VL - 16

JO - Journal of Integer Sequences

JF - Journal of Integer Sequences

IS - 7

ER -

On intervals (kn, (k + 1)n) containing a prime for all n > 1

Abstract

ASJC Scopus subject areas

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