Related papers: Fermat's Last Theorem: Algebra and Number Theory
We prove that several results in different areas of number theory such as the divergent series, summation of arithmetic functions, uniform distribution modulo one and summation over prime numbers which are currently considered to be…
The paper gives a unified and simple proof of both theorems and Cousin's theorem.
This paper provides a proof of a LLT-like test for Fermat numbers, based on the properties of Lucas Sequences and on the method of Lehmer.
In this paper we present theorems and applications of Wallis theorem related to trigonometric integrals.
In the paper we prove a new upper bound for Heilbronn's exponential sum and obtain some applications of our result to distribution of Fermat quotients.
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…
We give new proofs of some well-known results from Invariant Theorey using the Kempf-Ness theorem.
In this paper, we consider sums of generalized polygonal numbers with repeats, generalizing Fermat's polygonal number theorem which was proven by Cauchy. In particular, we obtain the minimal number of generalized $m$-gonal numbers required…
We study arithmetic distribution relations and the inverse function theorem in algebraic and arithmetic geometry, with an emphasis on versions that can be applied uniformly across families of varieties and maps. In particular, we prove two…
We survey facts mostly emerging from the seminal results of Alan Cobham obtained in the late sixties and early seventies. We do not attempt to be exhaustive but try instead to give some personal interpretations and some research directions.…
Many proofs of the Fundamental Theorem of Algebra, including various proofs based on the theory of analytic functions of a complex variable, are known. To the best of our knowledge, this proof is different from the existing ones.
In this paper, we consider some hybrid Diophantine equations of addition and multiplication. We first improve a result on new Hilbert-Waring problem. Then we consider the equation \begin{equation} \begin{cases} A+B=C ABC=D^n \end{cases}…
We survey recent applications of topology and singularity theory in the study of the algebraic complexity of concrete optimization problems in applied algebraic geometry and algebraic statistics.
The numerical analysis for the small amplitude motion of an elastic beam with internal damping is investigated in domain with moving ends. An efficient numerical method is constructed to solve this moving boundary problem. The stability and…
This is a survey on the finite basis problem for varieties of algebraic systems. Our exposition is in two directions: (i) We give numerous examples of varieties which are not finitely based. (ii) We give examples of important varieties with…
Using various results from extremal set theory (interpreted in the language of additive combinatorics), we prove an asyptotically sharp version of Freiman's theorem in F_2^n: if A in F_2^n is a set for which |A + A| <= K|A| then A is…
We will see that key concepts of number theory can be defined for arbitrary operations. We give a generalized distributivity for hyperoperations (usual arithmetic operations and operations going beyond exponentiation) and a generalization…
Fermat's Last Theorem (FLT) implies that the Frey curves do not exist. A proof of FLT independent of proved Taniyama-Shimura-Weil conjecture is presented.
We investigate some extremal problems in Fourier analysis and their connection to a problem in prime number theory. In particular, we improve the current bounds for the largest possible gap between consecutive primes assuming the Riemann…
Under integral restrictions on dilatations, it is proved existence theorems for the degenerate Beltrami equations with two characteristics and, in particular, to the Beltrami equations of the second type that play a great role in many…