On integers of the forms and

✍ Scribed by Yong-Gao Chen

• Publisher: Elsevier Science
• Year: 2007
• Tongue: English
• Weight: 143 KB
• Volume: 125
• Category: Article
• ISSN: 0022-314X
• DOI: 10.1016/j.jnt.2006.10.005

✦ Synopsis

In this paper we consider the integers of the forms k ± 2 n and k2 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 , k2 n + 1 and k2 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 , k2 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 , k2 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.