On the solutions of the equation 𝑥^{𝑚}+𝑦^{𝑚}-𝑧^{𝑚}=1 in a finite field

Abstract

Abstract. An explicit formula for the number of solutions of the equation in the title is given when a certain condition, depending only on m and the characteristic of the field, holds.

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

“…The present work has some connections with the investigation carried out by Ke and Kiechle in [10] since both have their origin in finite geometry. However, their results do not apply to our case.…”

“…Thus (qK, (qK!1)/n) is not circular. This shows that Theorem 1.1 does not follow from the results given in [10].…”

“…The reason is that the circularity condition assumed by them is not consistent with out assumption n"N, as the following argument shows. With out notation, a pair (qK, k) is circular (as defined in [10]) if the multiplicative subgroup satisfies " a#b 5 c#d"42 for all a, c 3GF(q)*, b, d3 GF(q) with aO c or bOd. Ke and Kiechle determined the number of solutions of the Fermat equation XL#½L#1"0 under the extra condition that (qK, (qK!1)/n) is circular.…”