Abstract :
In this paper we consider the integers of the forms k±2n and k2n±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−2n, k+2n, k2n+1 and k2n−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+2n, k+1+2n, k+2+2n, k+3+2n, k+4+2n, k2n+1, (k+1)2n+1, (k+2)2n+1, (k+3)2n+1 and (k+4)2n+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+2n, k+2+2n, k+4+2n, k+6+2n, k+8+2n, k2n+1, (k+2)2n+1, (k+4)2n+1, (k+6)2n+1 and (k+8)2n+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.
