Five consecutive positive odd numbers, none of which can be expressed as a sum of two prime powers

Abstract

In this paper, we prove that there is an arithmetic progression of positive odd numbers for each term M M of which none of five consecutive odd numbers M , M − 2 , M − 4 , M − 6 M, M-2, M-4, M-6 and M − 8 M-8 can be expressed in the form 2 n ± p α 2^n \pm p^\alpha , where p p is a prime and n , α n, \alpha are nonnegative integers.