Abstract Fermat’s last theorem appears not as a unique property of natural numbers but as the bottom line of extended possible issues involving larger dimensions and powers when observed from a natural vector space viewpoint. The fabric of this general Fermat’s theorem structure consists of a well-defined set of vectors associated with dimensional vector spaces and the Minkowski norms one can define there. Here, a special vector set is studied and named a Fermat surface. Besides, a connection between Fermat surfaces and hypercubes is unveiled.

Abstract We construct the galaxies of sequences of Toeplitz matrix solutions of the Diophantine equation Xn + Y n = Z n , n ≥ 3, linked to Pythagorean triples.