Sum of terms of recurrence sequences in the solution sets of generalized Pell equations

2024/03/27 by Pritam Kumar Bhoi, Bhoi, Pritam Kumar, Rudranarayan Padhy +3
Mathematics · #Advanced Optimization Algorithms Research

paper · pdf · doi:10.48550/arxiv.2403.18924

Abstract

Let (X_k)_{k≥ 1} and (Y_k)_{k≥ 1} be the sequence of X and Y-coordinates of the positive integer solutions (x, y) of the equation x² - dy² = t. In this paper we completely describe those recurrence sequences such that sums of two terms recurrence sequences in the solution sets of generalized Pell equations are infinitely many. Further, we give an upper bound for the number of such terms when there are only finitely many of them. This work is motivated by the recent paper Hajdu and Sebestyén (Int. J. Number Theory 18 (2022), 1605-1612).

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