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

abstract Let be the finite field of odd prime power order q, We find explicit expressions for the number of triples \-1, , +1 \ of consecutive non-zero squares in 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 \ -1, , +1, +2\ of square and non-squares over, when q is a power of 5. abstract