Related papers: General Elementary Direct Proof of Fermat's Last T…
New formulas for the construction of Pythagorean triples and generalizations to equations of higher powers. Application of formulas to some problems, in particular Fermat's equation with n=4.
The goal of this paper is to generalize the main results of [KM] and subsequent papers on metric Diophantine approximation with dependent quantities to the set-up of systems of linear forms. In particular, we establish `joint strong…
The paper presents a new generalized integer orthogonal transformation which consists of a well known orthogonal transform followed by stretching the basis vectors maintaining the asymptotic behavior of the number of integer solutions for…
We present a new, elementary, dynamical proof of the prime number theorem.
The paper gives a unified and simple proof of both theorems and Cousin's theorem.
This paper shows an elementary and direct proof of the Fundamental Theorem of Algebra, via Bolzano-Weierstrass Theorem on Minima and the Binomial Formula, that avoids: any root extraction other than the one used to define the modulus…
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…
It is an original method based on systems of prameters represented by reals which obey to an infinite descent (convergent sequences). We define calculus of quotients and they conduct quickly to a consequent result. Our own scepticism made…
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…
Our goal is to finally settle the persistent problem in Diophantine Approximation of finding best linear approximates. Classical results from the theory of continued fractions provide the solution for the special homogeneous case in the…
By applying inter-universal Teichm\"uller theory and its slight modification over the rational number field, we prove new Diophantine results towards effective abc inequalities and the generalized Fermat equations. For coprime integers $a,…
Let $\mathscr{P}_\mathbb{Q}=\{ \alpha^n \; : \; \alpha \in \mathbb{Q}, \; n \ge 2\}$ be the set of rational perfect powers, and let $S \subseteq \mathscr{P}_\mathbb{Q}$ be a finite subset. We prove the existence of a polynomial $f_S \in…
We present an elementary, $L^2,$ proof of Fedi\u{\i}'s theorem on arbitrary (e.g., infinite order) degeneracy and extensions. In particular, the proof allows and shows $C^\infty,$ Gevrey, and real analytic hypoellipticity, and allows the…
The first case of Fermat's Last Theorem for a prime exponent $p$ can sometimes be proved using the existence of local obstructions. In 1823, Sophie Germain has obtained an important result in this direction by establishing that, if $2p+1$…
Let $p \ge 5$ be a prime and let $\mathbb{Q}_{n,p}$ denote the $n$-th layer of the cyclotomic $\mathbb{Z}_p$-extension of $\mathbb{Q}$. We show that $\mathbb{Q}_{n,p}$ has no exceptional units. We use this to prove the effective asymptotic…
The main aim of the present paper is to represent an exact and simple proof for FLT by using properties of the algebra identities and linear algebra.
The aim of this short note is to present an elementary, self-contained, and direct proof for the classical Lebesgue decomposition theorem.
This is a detailed survey -- with rigorous and self-contained proofs -- of some of the basics of elementary combinatorics and algebra, including the properties of finite sums, binomial coefficients, permutations and determinants. It is…
Herbrand's Theorem is a fundamental result in mathematical logic which provides a reduction of first-order formulas satisfied by a universal class to formulas free of existential quantifiers. In this work, a simpler and self-contained…
We survey the classical results of the Dirichlet Approximation Theorem.