Consider the finite field $\mathbb{F}_q$. Schur (1916) proved that, given $n$, when the field is sufficient large, this equation,
$$x^n+y^n= z^n$$
always has a nontrivial solution.
What conditions does the number of solutions satisfy?
- I. Schur, Über die Kongruenz $x^{m} + y^{m} \equiv z^{m} \pmod{p}$, Jahresber. Deutschen Math. Verein. 25 (1916), 114–117.
Best Answer
There are some results and references in Lang, Cyclotomic Fields, 1.§6 (p. 22ff. in Cyclotomic Fields I and II, Combined Second Edition).
Let $V(d)$ be the Fermat curve of degree $d$. Theorem 6.1. The number of points of $V(d)$ (in affine space) is $q^2 - (q-1)\sum\chi^{a+b}(-1)J(\chi^a,\chi^b)$, the sum over integers $a,b$ with $0 < a,b < d$ and $a+b \not\equiv 0 \pmod{d}$ and $\chi$ the character such that $\chi(u) = \omega(u)^{(q-1)/d}$ with $\omega: \mathbf{F}_q \to \mu_{q-1}$ the Teichmüller character and the Jacobi sum $J(\chi_1,\chi_2) = -\frac{S(\chi_1)S(\chi_2)}{S(\chi_1\chi_2)}$ and the Gauß sum $S(\chi) = \sum_u\chi(u)\lambda(u)$ and $\lambda: \mathbf{F}_q \to \mu_p, \lambda(x) = \exp(2\pi i/p\mathrm{Tr}(x))$.