Triples and quadruples of consecutive squares or non-squares in a finite field

Abstract

Let $\F$ be the finite field of odd prime power order $q$, We find explicit expressions for the number of triples $\{\al-1,\al,\al+1 \}$ of consecutive non-zero squares in $\F$ and similarly for the number of triples of consecutive non-square elements. A key ingredient is the evaluation of Jacobsthal sums over general finite fields by Katre and Rajwade. This extends results of Monzingo(1985) to non-prime fields. Curiously, the same machinery alows the evaluation of the number of consecutive quadruples $\{\al -1, \al,\al+1, \al +2\}$ of square and non-squares over $\F$, when $q$ is a power of 5.