q-Congruences, with applications to supercongruences and the cyclic sieving phenomenon

We establish a supercongruence conjectured by Almkvist and Zudilin, by proving a corresponding $q$-supercongruence. Similar $q$-supercongruences are established for binomial coefficients and the Ap\'{e}ry numbers, by means of a general criterion involving higher derivatives at roots of unity. Our methods lead us to discover new examples of the cyclic sieving phenomenon, involving the $q$-Lucas numbers.Comment: Incorporated comments from referees. Accepted for publication in Int. J. Number Theor

A primality criterion based on a Lucas' congruence

Let $p$ be a prime. In 1878 \'{E}. Lucas proved that the congruence $${p-1\choose k}\equiv (-1)^k\pmod{p}$$ holds for any nonnegative integer $k\in\{0,1,\ldots,p-1\}$. The converse statement was given in Problem 1494 of {\it Mathematics Magazine} proposed in 1997 by E. Deutsch and I.M. Gessel. In this note we generalize this converse assertion by the following result: If $n>1$ and $q>1$ are integers such that $${n-1\choose k}\equiv (-1)^k \pmod{q}$$ for every integer $k\in\{0,1,\ldots, n-1\}$, then $q$ is a prime and $n$ is a power of $q$.Comment: 6 page