A note on integral matrices associated to finite fields (Q1102327)

Let \({\mathbb{F}}_ q\) be the finite field of q elements, \(q\equiv 1 (e)\), e a fixed integer, \(H_ f\) the unique subgroup of \({\mathbb{F}}^*_ q\) of order \(f=(q-1)/e\), g a generator of \({\mathbb{F}}^*_ q\), \(c_{ij}=card(H_ f\cap (g^ i+g^ jH_ f))\) (0\(\leq i,j\leq e-1)\) and \(c=c(q)=\min_{i,j}(c_{ij}).\) \textit{E. Snapper} [J. Algebra 97, 267-277 (1985; Zbl 0573.12007)] conjectured that \(\lim_{q\to \infty}c(q)=\infty\), where q ranges over primes \(\equiv 1 (mod e)\). - By using a well known lower bound for the number of solutions of \(ax^ e+by^ e=c\) over \({\mathbb{F}}_ q\), the author proves Snapper's conjecture where q ranges over prime powers\(\equiv 1 (mod e)\).