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

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. Bertók, L. Hajdu, I. Pink, and Z. Rábai, Linear combinations of prime powers in binary recurrence sequences, Int. J. Number Theory, 13(2):261–271, 2017].