Sieving Very Thin Sets of Primes, and Pratt Trees with Missing Primes

Kevin Ford

Type: Article

Publication Date: 2013-02-27

Citations: 3

DOI: https://doi.org/10.1093/imrn/rnt026

Abstract

Suppose that is a set of primes, such that for every ⁠, every prime factor of p−1 is also in ⁠. We apply a new sieve method to show that either contains all of the primes or the counting function of is O(x1−c) for some c>0, where c depends only on the smallest prime not in ⁠. Our proof makes use of results connected with Artin's primitive root conjecture.

Locations

International Mathematics Research Notices - View
arXiv (Cornell University) - View - PDF

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Works Cited by This (16)

Action	Title	Year	Authors
+	On the Normal Behavior of the Iterates Of some Arithmetic Functions	1990	Paul Erdős Andew Granvilie Carl Pomerance Claudia Spiro
+ PDF Chat	On Euler’s 𝜙-function	1907	R. D. Carmichael
+	On Artin's conjecture.	1967	C. Hooley
+ PDF Chat	The Lucas-Pratt primality tree	2007	Jonathan Bayless
+ PDF Chat	Note on Euler’s 𝜑-function	1922	R. D. Carmichael
+ PDF Chat	Prime Chains and Pratt Trees	2010	Kevin Ford Sergeĭ Konyagin Florian Luca
+	Introduction à la théorie analytique et probabiliste des nombres	2015	Gérald Tenenbaum
+	A remark on Artin's conjecture	1984	Rajiv Gupta M. Ram Murty
+	ARTIN'S CONJECTURE FOR PRIMITIVE ROOTS	1986	D. R. Heath‐Brown
+	A Curious Connection Between Fermat Numbers and Finite Groups	2002	Carrie E. Finch Lenny Jones