Affordable Access

Publisher Website

On integers of the forms [formula omitted]and [formula omitted]

Authors
Journal
Journal of Number Theory
0022-314X
Publisher
Elsevier
Publication Date
Volume
125
Issue
1
Identifiers
DOI: 10.1016/j.jnt.2006.10.005
Keywords
  • Covering Systems
  • Odd Numbers
  • Sums Of Prime Powers
Disciplines
  • Mathematics

Abstract

Abstract In this paper we consider the integers of the forms k ± 2 n and k 2 n ± 1 , which are ever focused by F. Cohen, P. Erdős, J.L. Selfridge, W. Sierpiński, etc. We establish a general theorem. As corollaries, we prove that (i) there exists an infinite arithmetic progression of positive odd numbers for each term k of which and any nonnegative integer n, each of four integers k − 2 n , k + 2 n , k 2 n + 1 and k 2 n − 1 has at least two distinct odd prime factors; (ii) there exists an infinite arithmetic progression of positive odd numbers for each term k of which and any nonnegative integer n, each of ten integers k + 2 n , k + 1 + 2 n , k + 2 + 2 n , k + 3 + 2 n , k + 4 + 2 n , k 2 n + 1 , ( k + 1 ) 2 n + 1 , ( k + 2 ) 2 n + 1 , ( k + 3 ) 2 n + 1 and ( k + 4 ) 2 n + 1 has at least two distinct odd prime factors; (iii) there exists an infinite arithmetic progression of positive odd numbers for each term k of which and any nonnegative integer n, each of ten integers k + 2 n , k + 2 + 2 n , k + 4 + 2 n , k + 6 + 2 n , k + 8 + 2 n , k 2 n + 1 , ( k + 2 ) 2 n + 1 , ( k + 4 ) 2 n + 1 , ( k + 6 ) 2 n + 1 and ( k + 8 ) 2 n + 1 has at least two distinct odd prime factors. Furthermore, we pose several related open problems in the introduction and three conjectures in the last section.

There are no comments yet on this publication. Be the first to share your thoughts.

Statistics

Seen <100 times
0 Comments

More articles like this

All the solutions of the equation [formula omitted...

on Discrete Mathematics Jan 01, 2008

[formula omitted]-full integers between successive...

on Indagationes Mathematicae Jan 01, 2011

On sets of integers, none of which divides the pro...

on European Journal of Combinator... Jan 01, 2011
More articles like this..