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

Abstract

Let (U n ) n≥0 be a nondegenerate binary recurrence sequence with positive discriminant. Let p 1 , . . . , p s be fixed prime numbers, b 1 , . . . , b s be fixed nonnegative integers, and a 1 , . . . , a t be positive integers. In this paper, under certain assumptions, we obtain a finiteness result for the solution of the Diophantine equation $$ {\alpha}_1{U}_{n1}+\cdots +{\alpha}_t{U}_{n1}={b}_1{p}_1^{z_1}+\cdots {b}_s{p}_s^{z_s}. $$ Moreover, we explicitly solve the equation F n1 + F n2 = 2 z1 + 3 z2 in nonnegative integers n 1, n 2, z 1, z 2 with z 2 ≥ z 1. The main tools used in this work are the lower bound for linear forms in logarithms and the Baker–Davenport reduction method. This work generalizes the recent papers [E. Mazumdar and S.S. Rout, Prime powers in sums of terms of binary recurrence sequences, arXiv:1610.02774] and [C. Bertok, L. Hajdu, I. Pink, and Z. Rabai, Linear combinations of prime powers in binary recurrence sequences, Int. J. Number Theory, 13(2):261–271, 2017].