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, square-free discriminant and $p$ be a fixed prime number. In this paper, we have shown the finiteness result for the solutions of the Diophantine equation $u_{n_1}+u_{n_2}+\cdots +u_{n_t}=p^z$ with some conditions on $n_i$ for all $1\le i\le t$. Moreover, we explicitly find all the powers of three which are sums of three balancing numbers using the lower bounds for linear forms in logarithms. Further, we use a variant of Baker-Davenport reduction method in Diophantine approximation due to Dujella and Pethő.

arXiv comment15 pages

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