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

Abstract

Let $\{U_n{\}}_{n\ge 0}$ be a non-degenerate binary recurrence sequence with positive discriminant. Let $\{p_1,\dots ,p_s\}$ be fixed prime numbers and $\{b_1,\dots ,b_s\}$ be fixed non-negative integers. In this paper, we obtain the finiteness result for the solution of the Diophantine equation $U_{n_1}+\cdots +U_{n_t}=b_1{p}_1^{z_1}+\cdots +b_s{p}_s^{z_s}$ under certain assumptions. Moreover, we explicitly solve the equation $F_{n_1}+F_{n_2}=2^{z_1}+3^{z_2}$, in non-negative integers $n_1,n_2,z_1,z_2$ with $z_2\ge z_1$. The main tools used in this work are the lower bound for linear forms in logarithms and the Baker-Davenport reduction method.

16 pages

References in corpus (1)

• Powers of two as sums of two $k-$Fibonacci numbers