Three consecutive integers with power of 5 mod 11 Let $(n - 1)$, $n$ and $(n + 1)$ be three consecutive integers, and $(n - 1)^5 \equiv n^5 \equiv (n + 1)^5 \equiv a \pmod{11}$, what are the possible values of $a$?
I know the facts that $3^5 \equiv 4^5 \equiv 5^5 \equiv 1 \pmod{11}$ and $6^5 \equiv 7^5 \equiv 8^5 \equiv -1 \pmod{11}$ are two solutions, however, is there a way to actually solve this equation?

There is sufficiently to check the numbers $-5,-4,\ldots,4,5$.
