• 1 University of Cambridge [UK]

Let $K (>1)$ and $k (>1)$ be given integers. In this paper we prove that $e_K(q)\equiv0 \mod k^{[m]}$ for infinitely many primes $q$, where $m=c_k\log\log q$ for a certain $c_k>0$ and $e_K(q)$ denotes the exponent of $K$ modulo $q$. In particular, $q\equiv1 \mod k$ for infinitely many primes $q$.