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Аннотация:Let N_q be the number of solutions to the equation
(x_1 + ⋯ + x_n)² = ax_1 ⋯ x_n
over the finite field F_q = F_{p^s}. Carlitz found formulas for N_q when n = 3 or 4. In an earlier paper, we found formulas for N_q when d = gcd(n - 2, q - 1) = 1 or 2 or 3 or 4; and when there exists an ℓ such that
p^ℓ ≡ -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 N_q when d = 8. We also simplify formulas for N_q when d = 4, p ≡ 1 (mod 4).