Linear combinations of prime powers in sums of terms of binary recurrence sequences

2016/12/18 by Meher, N. K., Rout, S. S.
#11J86 #FOS: Mathematics #Number Theory (math.NT) #Primary 11B37 #Secondary 11D72

paper · doi:10.48550/arxiv.1612.05869

Abstract

Let \ U_n\_{n ≥ 0} be a non-degenerate binary recurrence sequence with positive discriminant. Let \p₁,…, p_s\ be fixed prime numbers and \b₁,… ,b_s\ be fixed non-negative integers. In this paper, we obtain the finiteness result for the solution of the Diophantine equation U_n₁ + ⋯ + U_n_t = b₁ p₁^z₁ + ⋯+ b_s p_s^z_s under certain assumptions. Moreover, we explicitly solve the equation F_n1+ F_n2= 2^z₁ +3^z₂, in non-negative integers n₁, n₂, z₁, z₂ with z₂≥ z₁. The main tools used in this work are the lower bound for linear forms in logarithms and the Baker-Davenport reduction method.

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