ON THE NUMBER OF SOLUTIONS TO THE EQUATION (χ_1 + + χ_n)~m = aχ_1???χ_n OVER THE FINITE FIELD F_q FOR gcd(m - n, q - 1) = 7 AND gcd(m-n,q-l) = 14

摘要

Let N_q be the number of solutions to the equation (χ_1 ++χ_n)~m = aχ_1???χ_n over the finite field F_q = F_ps. L. Carlitz found formulas for N_q for m = 2, n = 3 or 4, and p > 2. In earlier papers, we found formulas for N_q when d = gcd(m n, q 1) = 1 or 2 or 3 or 4 or 6 or 8; and when there exists an ? such that d | (p~? + 1). In this paper, we find formulas for N_q when d = 7 or 14 and p = 2 or 4 (mod 7).