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On the generalized Sierpiński and Riesel numbers
Sobre los números de Sierpiński y Riesel generalizados
DOI:
https://doi.org/10.15446/bol.mat.v33n1.125921Palabras clave:
Brier numbers, covering systems, primitive prime divisors, Riesel numbers, Sierpiński numbers (en)números de Brier, sistemas de cubrimiento, divisores primos primitivos, números de Sierpiński, números de Riesel (es)
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In this paper we prove that for any fixed base b >= 2 there exist infinitely many positive integers k such that both kb^n + 1 and kb^n - 1 have at least two distinct prime factors for every positive integer n. In particular, this implies the existence of infinitely many integers that are simultaneously generalized Sierpinski and generalized Riesel numbers. Our approach uses covering systems following the ideas of Erdos and later constructions of Harrington. The argument is then completed using Zsigmondy's theorem on primitive divisors together with properties of multiplicative orders.
En este articulo se demuestra que, para toda base fija b >= 2, existen infinitos enteros positivos k tales que tanto k b^n + 1 como k b^n - 1 poseen al menos dos factores primos distintos para todo entero positivo n. En particular, esto implica la existencia de infinitos enteros que son simultaneamente numeros de Sierpinski generalizados y numeros de Riesel generalizados. Nuestro enfoque utiliza sistemas de cubrimiento, siguiendo las ideas de Erdos y construcciones posteriores de Harrington. El argumento se completa mediante el teorema de Zsigmondy sobre divisores primitivos, junto con propiedades de los ordenes multiplicativos.
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