Geodesic
On the number of solutions of the equation $(x_1+\dots+x_n)^2=ax_1\dots x_n$ in the finite field $\mathbb F_q$ for $\mathrm{gcd}(n-2,q-1)=7$ and for $\mathrm{gcd}(n-2,q-1)=14$
Čebyševskij sbornik, Tome 1 (2001) no. 1, pp. 5-14.

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Yu. N. Baulina. On the number of solutions of the equation $(x_1+\dots+x_n)^2=ax_1\dots x_n$ in the finite field $\mathbb F_q$ for $\mathrm{gcd}(n-2,q-1)=7$ and for $\mathrm{gcd}(n-2,q-1)=14$. Čebyševskij sbornik, Tome 1 (2001) no. 1, pp. 5-14. http://geodesic.mathdoc.fr/item/CHEB_2001_1_1_a0/

[1] Carlitz L., “The number of solutions of some equations in a finite field”, Portug. Math., 13:1 (1954), 25–31 | MR | Zbl

[2] Baulina Yu. N., O chisle reshenii uravneniya $(x_1+\dots+x_n)^2=ax_1\dots x_n$ v konechnom pole, Rukopis dep. v VINITI 4.05.2001, No 1148-V2001, Mos. ped. gos. un-t., M., 2001, 38 pp.

[3] Lidl R., Niderraiter G., Konechnye polya, «Mir», M., 1988 | Zbl

[4] Williams K. S., Hardy K., Spearman B. K., “Explicit evaluation of certain Eisenstein sums file”, Number Theory, ed. R. A. Mallin, de Gruyter, Berlin, 1990, 553–626 | MR