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On k-Pell numbers close to power of 2
Números de k-Pell cercanos a potencias de 2
DOI:
https://doi.org/10.15446/recolma.v58n1.117434Keywords:
Diophantine equations, k-Pell numbers, linear forms in logarithms, reduction method (en)Ecuaciones Diofantina, números k-Pell, formas lineales en logaritmos, metodo de reducción (es)
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For k ≥ 2, let (P(k)n)n≥2−k be the k-generalized Pell sequence which starts with 0, · · · , 0, 1 (k terms) and each term afterwards is given by the linear recurrence
P(k)n = 2P(k)n-1 + P(k)n-2 + · · · + P(k)n-k, for all n ≥ 2.
An integer n is said to be close to a positive integer m if n satisfies |n−m| < √m. In this paper, we solve the Diophantine inequality
|P(k)n − 2m| < 2m/2,
in positive unknowns k, n, and m.
Para k ≥ 2, sea (P(k)n)n≥2−k la k-sucesión generalizada de Pell que comienza en los valores 0, · · · , 0, 1 (k términos en total) y que satisface la relación de recurrencia
P(k)n = 2P(k)n-1 + P(k)n-2 + · · · + P(k)n-k, para todo n ≥ 2.
Un entero n se denomina cercano a otro entero m si n satisface |n−m| < √m. En este artículo se resuelve la desigualdad Diofantina
|P(k)n − 2m| < 2m/2,
para las indeterminadas enteras k, n, y m
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