Related papers: Fermat's Last Theorem: Algebra, Geometry, and Numb…

In this article we present a generalization of a Leibniz's geometrical theorem and an application of it.

In this thesis, written in Italian, some original results are presented: two new optical invariants, similar to that of Lagrange, the generalization of the third order Luneburg's aberrations formulae and the detailed proof of the way they…

We demonstrate how one can see quantization of geometry, and quantum algebraic structure in supersymmetric gauge theory.

We will be presenting two theorems in this paper. The first theorem, which is a new result, is about the non-existence of integer solutions of the cubic diophantine equation. In the proof of this theorem we have used some known results from…

In this text, we wish to provide the reader with a short guide to recent works on the theory of dilatations in Commutative Algebra and Algebraic Geometry. These works fall naturally into two categories: one emphasises foundational and…

The divisibility restrictions in the famous equation a n+bn=cn in Fermat Last Theorem (FLT, 1637) is analyzed how it selects out many triples to be Fermat triple (i.e. solutions) if n greater than 2, decreasing the cardinality of Fermat…

A study of Sophie Germain's extensive manuscripts on Fermat's Last Theorem calls for a reassessment of her work in number theory. There is much in these manuscripts beyond the single theorem for Case 1 for which she is known from a…

We show that a single special separation theorem (namely, a consequence of the geometric form of the Hahn-Banach theorem) can be used to prove Farkas type theorems, existence theorems for numerical quadrature with positive coefficients, and…

We prove Roth type theorems in finite groups. Our main tool is the Triangle Removal Lemma of Ruzsa and Szemer\'edi.

We give a new proof of the fundamental theorem of algebra. It is entirely elementary, focused on using long division to its fullest extent. Further, the method quickly recovers a more general version of the theorem recently obtained by…

Hilbert's Foundations of Geometry was perhaps one of the most influential works of geometry in the 20th century and its axiomatics was the first systematic attempt to clear up the logical gaps of the Elements. But does it have gaps of its…

In this note we list a number of open problems in the fields of number theory, combinatorics, and representation theory: algebraic functions with Fermat property; power product expansion of the generating function for the partition…

This paper is a sequel to [3]. We formulate a natural algebraic geometry conjecture, give some of its number theoretic and analytical consequences, and show that those can be used to get further advances in wave turbulence theory.

We prove some results on the border of Ramsey theory (finite partition calculus) and model theory. Also a beginning of classification theory of finite models in undertaken.

Intended for mathematical physicists interested in applications of the division algebras to physics, this article highlights some of their more elegant properties with connections to the theories of Galois fields and quadratic residues.

We review some recent applications of machine learning to algebraic geometry and physics. Since problems in algebraic geometry can typically be reformulated as mappings between tensors, this makes them particularly amenable to supervised…

This paper initiates a novel research direction in the theory of Diophantine equations: define an appropriate version of the equation's size, order all polynomial Diophantine equations starting from the smallest ones, and then solve the…

Partially or totally unsolved questions in number theory and geometry especially, such as coloration problems, elementary geometric conjectures, partitions, generalized periods of a number, length of a generalized period, arithmetic and…

In this paper we provide an application to the Neumann problem of a recent three critical points theorem.

We survey some recent applications of machine learning to problems in geometry and theoretical physics. Pure mathematical data has been compiled over the last few decades by the community and experiments in supervised, semi-supervised and…