Bulletin of the Australian Mathematical Society

Research Article

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

XUE-GONG SUNa1a2 and JIN-HUI FANGa3

a1 Department of Mathematics, Nanjing Normal University, Nanjing 210097, People’s Republic of China

a2 Department of Mathematics and Science, Huai Hai Institute of Technology, Lian Yun Gang 222005, Jiangsu, People’s Republic of China (email: fangjinhui1114@163.com)

a3 Department of Mathematics, Nanjing Normal University, Nanjing 210097, People’s Republic of China

Abstract

Erdős 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.

(Received February 18 2008)

2000 Mathematics subject classification

  • 11A07;
  • 11B25;
  • 11P32

Keywords and phrases

  • asymptotic density;
  • covering systems;
  • arithmetic progressions

Footnotes

This work was supported by the National Natural Science Foundation of China Grant Nos. 10771103 and 10801075.