ON THE DENSITY OF INTEGERS OF THE FORM (p−1)2−n IN ARITHMETIC PROGRESSIONS

Author:

SUN XUE-GONG,FANG JIN-HUI

Abstract

AbstractErdős and Odlyzko proved that odd integers k such that k2n+1 is prime for some positive integer n have a positive lower density. In this paper, we characterize all arithmetic progressions in which natural numbers that can be expressed in the form (p−1)2n (where p is a prime number) have a positive proportion. We also prove that an arithmetic progression consisting of odd numbers can be obtained from a covering system if and only if those integers in such a progression which can be expressed in the form (p−1)2n have an asymptotic density of zero.

Publisher

Cambridge University Press (CUP)

Subject

General Mathematics

Cited by 5 articles. 订阅此论文施引文献 订阅此论文施引文献,注册后可以免费订阅5篇论文的施引文献,订阅后可以查看论文全部施引文献

1. Tight upper and lower bounds for the reciprocal sum of Proth primes;The Ramanujan Journal;2022-01-17

2. On a problem of Romanoff type;Acta Arithmetica;2022

3. Primes of the form $${kM^n+n}$$ k M n + n;Acta Mathematica Hungarica;2019-01-24

4. ON THE INTEGERS OF THE FORM $p+b$;Taiwanese Journal of Mathematics;2014-09-01

5. An Extension of the Dirichlet Density for Sets of Gaussian Integers;Canadian Mathematical Bulletin;2013-03-01

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