In a 1975 paper `Not Every Number is the Sum or Difference of Two Prime Powers', Cohen and Selfridge use covering congruences to prove their Theorem 1, which states that there exists an arithmetic progression of odd numbers which are neither the sum nor difference of a power of two and a prime.
The paper is available here:
As I understand their explanation, they are claiming that if $M$ satisfies a list of congruences (given on the left-hand side of the table on page 2 of the paper), then for any $n$, there will be a prime $p_i$ which is a factor of $M+2^n$ and a prime $p_j$ which is a factor of $M-2^n$, and then they deduce that $M+2^n$ and $M-2^n$ are not prime.
My difficulty stems from the fact that while I can see that their conclusion that there is a $p_j$ which is a factor of $M-2^n$ is clearly justified, I cannot see how to elimiate the possibility that $M-2^n$ might actually be equal to $p_j$.
I have looked back at the paper of Erdős:
which is referenced by Cohen and Selfridge, but have not found that it solves my problem.
To a certain extent, my query might be somewhat academic, since, in Theorem 2 of the Cohen and Selfridge paper, they extend their method and their coverings to prove the existence of two distinct prime factors, but I am curious as to whether I have missed something obvious in Theorem 1.