Related papers: Fermat's Last Theorem: Algebra and Number Theory

In this paper, we generalized the known Laplace-transform final-value theorem. From our conclusion, one can deduce the existing results in [1, 3, 12]. By using final value theorem, we give a new proof that Caputo fractional differential…

A proof is given to the Dixmier's 5'th problem for the Weyl algebra.

Can any element in a sufficiently large finite field be represented as a sum of two $d$th powers in the field? In this article, we recount some of the history of this problem, touching on cyclotomy, Fermat's last theorem, and diagonal…

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 give a simple direct proof of Fermat's two squares theorem. Our argument uses no intricate notions or ideas; one might say that it is a proof by careful bookkeeping. As such, the proof may be particularly easy to comprehend by students…

We give a reformulation of the Lehmer conjecture about algebraic integers in terms of a simple counting problem modulo p.

The aim of this work is to develop general optimization methods for finite difference schemes used to approximate linear differential equations. The specific case of the transport equation is exposed. In particular, the minimization of the…

In this paper, we derive Taylor's theorem for beta-fractional derivative. We also investigate some new properties of Taylor's theorem and some useful related theorems for this derivative. We extend some recent and classical integral…

Let $A$ be a sufficiently dense subset of a finite field $\mathbb F_q$ or a finite, cyclic ring $\mathbb Z/ N\mathbb Z$. Assuming that $q$ and $N$ have no small prime divisors, we show that generalised Fermat equations have the expected…

This study use some philological and historical means in order to understand Fermat's way of thinking.

We consider the Diophantine equation $x^4 + y^4 - w^2 = n$ for $n \in \mathbb{Z}$, which is related to near misses for the quartic case of Fermat's Last Theorem. For certain $n$ we show that the set of solutions is infinite, or more…

Some symmetry problems are formulated and solved. New simple proofs are given for the earlier studied symmetry problems.

In this paper, we show the new fixed point theorem in metric spaces. Furthermore, for this fixed point theorem, we apply to the Collatz conjecture.

We give a new proof of Lucas' Theorem in elementary number theory.

An extension to the factorisation principle as suggested by Fermat is presented.We start from a symmetry of natural numbers and obtain the factorisation principle therefrom.Later it is extended further to test the primality of any natural…

In this paper, we study the integer solutions of a family of Fermat-type equations of signature $(2, 2n, n)$, $Cx^2 + q^ky^{2n} = z^n$. We provide an algorithmically testable set of conditions which, if satisfied, imply the existence of a…

Several quantum systems have been used in the last few years to extend supersymmetry. In this paper we show all this systems fit into the picture of what we call "Number Operator Algebras".

In this paper, we have found that some certain Fermat-type shift and difference equations have the meromorphic solutions generated by Riccati type functions. Also we have solved the open problems posed by Liu and Yang (A note on meromorphic…

This is an old article of 2000. Its aim is to illustrate how a Lie-theoretic result of Zelmanov enables one to treat various problems in group theory.

We show that $\lambda$-symmetries can be algorithmically obtained by using the Jacobi last multiplier. Several examples are provided.