ON A PROBLEM OF CARLITZ
Ioulia N. Baoulina · International Journal of Number Theory · 2005
Let Nq be the number of solutions to the equation [Formula: see text] over the finite field [Formula: see text]. Carlitz found formulas for Nq when n = 3 or 4. In an earlier paper, we found formulas for Nq when d = gcd (n - 2, q - 1) = 1 or 2 or 3 or 4; and when there exists an l such that pl ≡ -1 (mod d). In another paper the cases d = 7 or 14, p ≡ 2 or 4 (mod 7) were considered. In this paper, we find formulas for Nq when d = 8. We also simplify formulas for Nq when d = 4, p ≡ 1 (mod 4).