My exercise sheet requires me to look at the following. I am looking for an elementary solution to (all polynomials have integer coefficients):
Do there exist cases of a>1 and M integers with a dividing M such that $(x+1)^M-2=0\mod(x^a-1,M)$?
So far I can prove no such cases exist for $a=2,3,4$ but can't make any further progress. Any help would much be appreciated!