Why does $x^{15}+x^{14}+1$ detect all errors at most 32768 bits apart?

Question

Reference Question from Forouzon Book Computer Network. Find the status of the following generator related to two isolated, single-bit errors. $$x^{15} + x^{14} + 1$$ Answer given : This polynomial cannot divide any error of type $x^t + 1$ if t is less than 32,768. This means that a codeword with two isolated errors that are next to each other or up to 32,768 bits apart can be detected by this generator. Can some one please help me why is this true...? Or even how should I have an in general approach to analyze such questions...? (Sorry if this is very silly, but I am unable to get a grasp of how the reason works...!)

Answer 1

Your polynomial is primitive, which means that the order of $x$ modulo your polynomial is exactly $2^{15}-1$. In particular, $x^a \not\equiv 1$ modulo your polynomial for all $1 \leq a \leq 2^{15}-2$, which means that your polynomial doesn't divide $x^a-1$ for this range of $a$.

(This also means that your polynomial should divide $x^{2^{15}-1}-1$, in contrast to what the question claims.)