You might like to look at some of the answers to Buzzard's question here: mathoverflow.net/questions/5323/… (Why does $2^n+9262111$ not qualify as a covering congruence argument which applies in your case?) More seriously, in order to avoid a repeat of the low quality answers to that question, perhaps you can make your question more precise and ask what you really want to know, which seems to be something like "is there a known and accepted way to make conjectures about these questions which avoids all the pitfalls I just mentioned (