Related papers: A new proof of Lucas' Theorem
We suggest an alternative proof of a theorem due to Lambek and Moser using a perceptible model.
In this paper we show an index theorem for gerbes
We present a linearity theorem for a proof language of intuitionistic multiplicative additive linear logic, incorporating addition and scalar multiplication. The proofs in this language are linear in the algebraic sense. This work is part…
Taylor's theorem (and its variants) is widely used in several areas of mathematical analysis, including numerical analysis, functional analysis, and partial differential equations. This article explains how Taylor's theorem in its most…
In this paper, by presenting bi-periodic Lucas numbers as a binomial sum, we introduce the bi-periodic incomplete Lucas numbers. After that, by using the bi-periodic incomplete Lucas numbers, we derive the recurrence relation and the…
We give a brief historical overview of the famous Pythagoras' theorem and Pythagoras. We present a simple proof of the result and dicsuss some extensions. We follow \cite{thales}, \cite{wiki} and \cite{wiki2} for the historical comments and…
The paper contains an alternative proof of M. Kontsevich Formality Theorem.
A simple proof of the celebrated theorem of Lee and Yang is attempted in this short note.
We provide a new simple and transparent proof of the version of Kummer's test given in [Tong, J. (1994). Amer. Math. Monthly. 101(5): 450--452]. Our proof is based on an application of a Hardy--Littlewood Tauberian theorem.
In this work, we prove that many Ap\'ery-like sequences arising from modular forms satisfy the Lucas congruences modulo any prime. As an implication, we completely affirm four conjectural Lucas congruences that were recently posed by S.…
The assumptions needed to prove Cox's Theorem are discussed and examined. Various sets of assumptions under which a Cox-style theorem can be proved are provided, although all are rather strong and, arguably, not natural.
In this note we provide a new proof of the Tikhonov theorem for the infinite time interval and discuss some of its applications.
Let $\Omega(n)$ denote the number of prime factors of $n$. We show that for any bounded $f\colon\mathbb{N}\to\mathbb{C}$ one has \[ \frac{1}{N}\sum_{n=1}^N\, f(\Omega(n)+1)=\frac{1}{N}\sum_{n=1}^N\, f(\Omega(n))+\mathrm{o}_{N\to\infty}(1).…
This paper contains a new elementary proof of the Fundamental Theorem of Calculus for the Lebesgue integral. The hardest part of our proof simply concerns the convergence in ${\rm L}^1$ of a certain sequence of step functions, and we prove…
The generalized Lucas numbers are polynomials in two variables with nonnegative integer coefficients. Lucas versions of some combinatorial numbers with known formulas in terms of quotient and products of nonnegative integers have been…
We prove a generalization of Lopes's theorem, that is, of the converse of Brolin's theorem.
The main purpose of this paper is to establish some new properties of Horadam numbers in terms of binomial sums. By that, we can obtain these special numbers in a new and direct way. Moreover, some connections between Horadam and…
This is an exposition of Gauss's proof of Descartes's rule of signs.
We prove Sklar's theorem in infinite dimensions via a topological argument and the notion of inverse systems.
We give four new proofs of the directed version of Brook's Theorem and an NP-completeness result.