Consecutive power residues or nonresidues

Abstract:

For any positive integers $k$ and $l$, A. Brauer [1] has shown that there exists a number $z(k,l)$ such that, for any prime number $p > z(k,l)$, a sequence of $l$ consecutive numbers occurs in at least one $k$th-power class modulo $p$. For particular $k$ and $l$, one is sometimes able to find a least bound, $\Lambda *(k,l)$, before, or at which, the first member of such a sequence must appear. In this paper, we describe a method used to compute $\Lambda *(8,2)$ and $\Lambda *(3,3)$.