On integers of the forms k±n2 and kn2±1

In this paper we consider the integers of the forms k±n2 and kn2±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−n2, k+n2, kn2+1 and kn2−1 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+n2, k+1+n2, k+2+n2, k+3+n2, k+4+n2, kn2+1, (k+1)n2+1, (k+2)n2+1, (k+3)n2+1 and (k+4)n2+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+n2, k+2+n2, k+4+n2, k+6+n2, k+8+n2, kn2+1, (k+2)n2+1, (k+4)n2+1, (k+6)n2+1 and (k+8)n2+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.