# 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.

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