Mathematical Framework and Computational Results Associated with Rational Points on Fermat’s Surfaces in Minkowski’s (N+1) - Dimensional Spaces and Extended Fermat’s Last Theorem

Abstract

Despite the elaborate Wiles demonstration, Fermat’s last theorem still attracts researchers to this aspect of number theory. This study has been structured in two parts. The first one describes the Minkowski natural space basis, where the discussion of Fermat’s theorem and the extensions to higher dimensional spacesss. In the second, many examples of N-dimensional vectors obeying a Fermat-like rule for various powers are presented and discussed. The Fermat last theorem, defined in (2+1)-dimensional Minkowski spaces, is discussed and extended in natural and rational Mikowski’s spaces. Several pieces of computational interest are given, with many practical examples. A definition of Fermat vector order, Fermat surfaces, and Fermat surface radius is given. Several conjectures are discussed, among them the existence of a Fermat theorem in (3+1)-dimensional Minkowski spaces.