A conjetura de Erdös-Straus e generalizações

Abstract

Nesta dissertação serão apresentados os principais resultados relacionados com a conjetura de Erd}os-Straus, entre os quais o teorema de Mordell. Associada à conjetura está uma equação diofantina, que iguala uma fração de numerador n = 4 e denominador m inteiro maior que 1, à soma de três frações unitárias. Será também analisado o número de soluções desta equação e vamos também verificar computacionalmente a validade da conjetura para m _ 109. Finalmente, iremos ver generalizações da conjetura para qualquer n e para quando, para além de somas, podemos também ter subtrações de frações unitárias. Ambas as generalizações foram propostas por André Schinzel.

This dissertation will present the main results related to the Erd}os- Straus conjecture, including the Mordell's theorem. Associated to the conjecture is a diophantine equation, that say that a fraction with numerator n = 4 and denominator m integer greater than 1 is equal to the sum of three unit fractions. It will be also analyzed the number of solutions of this equation and we also computationally verify the validity of the conjecture for m 109. Finally, we will see generalizations of the conjecture for any n and when, in addition to sums, we also have subtractions of unit fractions. Both generalizations were proposed by Andr é Schinzel.

This dissertation will present the main results related to the Erd}os- Straus conjecture, including the Mordell's theorem. Associated to the conjecture is a diophantine equation, that say that a fraction with numerator n = 4 and denominator m integer greater than 1 is equal to the sum of three unit fractions. It will be also analyzed the number of solutions of this equation and we also computationally verify the validity of the conjecture for m _ 109. Finally, we will see generalizations of the conjecture for any n and when, in addition to sums, we also have subtractions of unit fractions. Both generalizations were proposed by André Schinzel.