Fermat Curves over Finite Fields and Cyclic Subsets in High-Dimensional Projective Spaces

Let GF(q) be the Galois field of orderq=ph, and letm≥3 be an integer. An explicit formula for the number of GF(qm)-rational points of the Fermat curveXn+Yn+Zn=0 is given whenndivides (qm−1)/(q−1) andpis sufficiently large with respect to (qm−1)/(n(q−1)).