ON THE NUMBER OF SOLUTIONS TO THE EQUATION (x1 + ⋯ + xn)2 = ax1 ⋯ xn IN A FINITE FIELD

Abstract

Let Nq be the number of solutions to the equation [Formula: see text] over the finite field 𝔽q = 𝔽ps. L. 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 our other paper, the cases d = 7 or 14, p ≡ 2 or 4 (mod 7) were considered. Recently, we obtained formulas for Nq when d = 8. In this paper, we find formulas for Nq when d = 2t, t ≥ 4, p ≡ 3 or 5 (mod 8) or p ≡ 9 (mod 16).