Related papers: Fermat's Last Theorem: Algebra and Number Theory
In this paper, we prove several generalizations and applications of a fixed point theorem. This theorem is used to prove the existence and uniqueness of solutions of the linear sparse matrix problem considered.
Classical applications of Galois theory concern algebraic numbers and algebraic functions. Still, the night before his duel, Galois wrote that his last mathematical thoughts had been directed toward applying his "theory of ambiguity to…
In this paper we mainly study the existence and the form of entire solutions with finite order for the following system of Fermat-type difference and partial differential-difference equations $$\begin{cases} f_1(z)^2+(\Delta_cf_2(z))^2=1\cr…
The non-zero integer solution set is derived for C^n = A^n + B^n. The non-zero integer solution set for n = 2 is [C - (a + b)]^2 = 2ab. The variables a and b equal (C - A) and (C - B) respectively and are nonzero integer factors of 2M^2…
We introduce a new framework called linear algebraic number theory (LANT) that reformulates the number-theoretic problem as a regression model and solves it using matrix algebra. This framework restricts all computations to log space,…
Finite difference schemes are here solved by means of a linear matrix equation. The theoretical study of the related algebraic system is exposed, and enables us to minimize the error due to a finite difference approximation.
The Modular Group provides simple proofs of Fermat's representations: X^2+Y^2 for primes congruent to 1 (mod 4) and by X^2+3Y^2 for primes congruent to 1 (mod 3)
We prove the Nagata compactification theorem for any separated map of finite type between quasi-compact and quasi-separated algebraic spaces, generalizing earlier results of Raoult. Along the way we also prove (and use) absolute noetherian…
The idea of generating integrals analogous to generating functions is first introduced in this paper. A new proof of the well-known Finite Harmonic Series Theorem in Analysis and Analytical Number Theory is then obtained by the method of…
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…
We will give several reduction theorems for Noether's problem.
We give here a new proof of a Tauberian Theorem of complex Laplace transform using the Theory of measure and theory of function with bounded variations. However we deduce the simple proof of Prime Number Theorem.
In this article we determine several theorems and methods for solving linear congruences and systems of linear congruences, and we find the number of distinct solutions. Many examples of solving congruences are given.
The first part is expository: it explains how finite fields may be used to prove theorems on infinite fields by a reduction mod p process. The second part gives a variant of P.Smith's fixed point theorem which applies in any characteristic.
In a recent paper, the first author provided some lower bounds to solutions of the equations of Fermat and Catalan, based on local power series developments at the ramified prime of a prime cyclotomic extension. Although both equations have…
Recent developments of Baxter algebras have lead to applications to combinatorics, number theory and mathematical physics. We relate Baxter algebras to Stirling numbers of the first kind and the second kind, partitions and multinomial…
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…
Ordinary differential equations have an arithmetic analogue in which functions are replaced by numbers and the derivation operator is replaced by a Fermat quotient operator. In this survey we explain the main motivations, constructions,…
The ABC conjecture implies many conjectures and theorems in number theory, including the celebrated Fermat's Last Theorem. Mason-Stothers Theorem is a function field analogue of the ABC conjecture that admits a much more elementary proof…
In this lecture notes we try to familiarize the audience with the theory of Bernoulli polynomials; we study their properties, and we give, with proofs and references, some of the most relevant results related to them. Several applications…