Abstract
Let N q be the number of solutions to the equation
over the finite field \( \mathbb{F} \) q = \( \mathbb{F} \) p s . 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 ℓ such that d|(pℓ + 1). In this paper, we find formulas for N q when d = 7 or 14 and p ≡ 2 or 4 (mod 7).
2000 Mathematics Subject Classification
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References
Yu. N. Baulina, On the number of solutions of the equation (x1+⋯+xn)2 = ax1⋯xn in the finite field \( \mathbb{F} \)q for gcd(n —2, q—1) = 7 and for gcd(n —2, q—1) = 14 (Russian), Chebyshevskiῐ Sb. 1 (2001) 5–14.
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Baoulina, I. (2009). On the Number of Solutions to the Equation (x1 + ⋯ + x n )m = ax1 ⋯ x n over the Finite Field \( \mathbb{F} \) q for gcd(m − n, q − 1) = 7 and gcd(m − n, q − 1) = 14. In: Adhikari, S.D., Ramakrishnan, B. (eds) Number Theory and Applications. Hindustan Book Agency, Gurgaon. https://doi.org/10.1007/978-93-86279-46-0_2
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DOI: https://doi.org/10.1007/978-93-86279-46-0_2
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