Artículos con la etiqueta ‘teoría de los números’

A Friendly Intro to Sieves with a Look Towards Recent Progress on the Twin Primes Conjecture

Por • 30 ene, 2014 • Category: Educacion

This is an extension and background to a talk I gave on 9 October 2013 to the Brown Graduate Student Seminar, called `A friendly intro to sieves with a look towards recent progress on the twin primes conjecture.’ During the talk, I mention several sieves, some with a lot of detail and some with very little detail. I also discuss several results and built upon many sources. I’ll provide missing details and/or sources for additional reading here.

On transcendental numbers

Por • 1 ene, 2014 • Category: Opinion

Transcendental numbers play an important role in many areas of science. This paper contains a short survey on transcendental numbers and some relations among them. New inequalities for transcendental numbers are stated in Section 2 and proved in Section 4. Also, in relationship with these topics, we study the exponential function axioms related to the Yang-Baxter equation.

Numbers as functions

Por • 19 dic, 2013 • Category: Crítica

In this survey I discuss A. Buium’s theory of “differential equations in the p-adic direction” ([Bu05]) and its interrelations with “geometry over fields with one element”, on the background of various approaches to p-adic models in theoretical physics.

Husserl, Cantor & Hilbert: La Grande Crise des Fondements Mathematiques du XIXeme Siecle

Por • 13 nov, 2013 • Category: Educacion

Three thinkers of the 19th century revolutionized the science of logic, mathematics, and philosophy. Edmund Husserl (1859-1938), mathematician and a disciple of Karl Weierstrass, made an immense contribution to the theory of human thought. The paper offers a complex analysis of Husserl’s mathematical writings covering calculus of variations, differential geometry, and theory of numbers which laid the ground for his later phenomenological breakthrough. Georg Cantor (1845-1818), the creator of set theory, was a mathematician who changed the mathematical thinking per se. By analyzing the philosophy of set theory this paper shows how was it possible (by introducing into mathematics what philosophers call ‘the subject’). Set theory happened to be the most radical answer to the crisis of foundations. David Hilbert (1862-1943), facing the same foundational crisis, came up with his axiomatic method, indeed a minimalist program whose roots can be traced back to Descartes and Cauchy. Bringing together these three key authors, the paper is the first attempt to analyze how the united efforts of philosophy and mathematics helped to dissolve the epistemological crisis of the 19th century.

Arithmetic properties of Apéry-like numbers

Por • 20 oct, 2013 • Category: Educacion

We provide lower bounds for p-adic valuations of multisums of factorial ratios which satisfy an Ap\’ery-like recurrence relation: these include Ap\’ery, Domb, Franel numbers, the numbers of abelian squares over a finite alphabet, and constant terms of powers of certain Laurent polynomials. In particular, we prove Beukers’ conjectures on the p-adic valuation of Ap\’ery numbers. Furthermore, we give an effective criterion for a sequence of factorial ratios to satisfy the p-Lucas property for almost all primes p.

Preventing Exceptions to Robins InEquality

Por • 19 ago, 2013 • Category: Crítica

For sufficiently large n Ramanujan gave a sufficient condition for the truth Robin’s InEquality $X(n):=\frac{\sigma(n)}{n\ln\ln n}

Models of true arithmetic are integer parts of nice real closed fields

Por • 7 ago, 2013 • Category: Educacion

Exploring further the connection between exponentiation on real closed fields and the existence of an integer part modelling strong fragments of arithmetic, we demonstrate that each model of true arithmetic is an integer part of an exponential real closed field that is elementary equivalent to the reals with exponentiation.
arXiv:1307.6595v1 [math.LO]

El concepto de número en G.Frege

Por • 30 jul, 2013 • Category: Educacion

El objetivo de este trabajo es delinear el camino que Frege siguió hasta llegar a la definición del número. El material que se presenta ha sido organizado de tal manera que eventualmente se llegue a comprender la respuesta a la pregunta:¿ qué es lo que significan realmente los números? El trabajo ha sido dividido en seis parágrafos. El primero se introduce los tres principios que Frege utilizó a lo largo de sus investigaciones. En los parágrafos segundo, tercero y cuarto se ha presentado el aparato conceptual de Frege. El quinto se presenta lo que Frege considera el análisis adecuado de las sentencias en que intervienen los numerales y finalmente en el sexto se presenta la definición canónica de número.

Metric number theory, lacunary series and systems of dilated functions

Por • 20 jul, 2013 • Category: Leyes

By a classical result of Weyl, for any increasing sequence $(n_k)_{k \geq 1}$ of integers the sequence of fractional parts $(\{n_k x\})_{k \geq 1}$ is uniformly distributed modulo 1 for almost all $x \in [0,1]$. Except for a few special cases, e.g. when $n_k=k, k \geq 1$, the exceptional set cannot be described explicitly. The exact asymptotic order of the discrepancy of $(\{n_k x\})_{k \geq 1}$ is only known in a few special cases, for example when $(n_k)_{k \geq 1}$ is a (Hadamard) lacunary sequence, that is when $n_{k+1}/n_k \geq q > 1, k \geq 1$. In this case of quickly increasing $(n_k)_{k \geq 1}$ the system $(\{n_k x\})_{k \geq 1}$ (or, more general, $(f(n_k x))_{k \geq 1}$ for a 1-periodic function $f$) shows many asymptotic properties which are typical for the behavior of systems of \emph{independent} random variables. Precise results depend on a fascinating interplay between analytic, probabilistic and number-theoretic phenomena.

A Study of Kummer’s Proof of Fermat’s Last Theorem for Regular Primes

Por • 19 jul, 2013 • Category: Educacion

We study Kummer’s approach towards proving the Fermat’s last Theorem for regular primes. Some basic algebraic prerequisites are also discussed in this report, and also a brief history of the problem is mentioned. We review among other things the Class number formula, and use this formula to conclude our study.