\(k\)-Fibonacci powers as sums of powers of some fixed primes (Q2664017)

For a fixed integer \( k\ge 2 \), the sequence of \textit{\( k \)-generalized Fibonacci numbers}, \( \{F_n^{(k)}\}_{n\ge -(2-k)} \), is defined by the linear recurrence: \begin{align*} F_{n}^{(k)}=F_{n-1}^{(k)}+ \cdots+ F_{n-k}^{(k)} \quad \text{for all} \quad n\ge2, \end{align*} with the initial conditions: \( F_{i}^{(k)}=0 \), for \( i=2-k, \ldots, 0 \), and \( F_{1}^{(k)}=1 \). When \( k=2 \), this sequence coincides with the classical \textit{Fibonacci sequence}. When \( k=3 \), it coincides with the \textit{Tribonacci sequence}, and so on. Let \( S=\{p_1, \ldots, p_t\} \) be a fixed set of primes arranged in increasing order. In the paper under review, the authors study the integer solutions \( (k,n,s,a_1, \ldots, a_t )\) of the exponential Diophantine equation: \begin{align*} \left(F_{n}^{(k)}\right)^{s}=p_1^{a_1}+ \cdots+ p_t^{a_t}, \end{align*} where \( a_i \) are nonnegative integers such that \( \max\{a_i: 1\le i\le t\}=a_t \). The main result of the authors in this paper is the following. Theorem 1. Let \( p_t\ge 3 \). If \( (k,n,s,a_1, \ldots, a_t) \) is a positive integer solution of the equation (1) with \( n\ge k+2 \) and \( n> C_1(p_t) \), then \begin{align*} \max\{n,s,a_t\}< \dfrac{2\cdot 10^{80}\log^{17}pt}{\log^{5}\delta}, \end{align*} where \( \delta = p_t/p_{t-1} \), with \( p_0=1 \) in the case \( t=1 \). Furthermore, for \( k\ge 2, ~n\ge 3 \), and \( s\ge 1 \), as an application of Theorem 1, the authors find all solutions of the exponential Diophantine equation \begin{align*} \left(F_{n}^{(k)}\right)^{s}=2^{a}+3^{b}+5^{c}, \quad \text{where} \quad 0\le \max\{a,b\} \le c. \end{align*} To prove their results, the authors use a clever combination of techniques in number theory, the usual properties of the \( k \)-generalized Fibonacci sequence, the theory of nonzero linear forms in logarithms of algebraic numbers, as well as reduction techniques involving the theory of continued fractions. All computations are done with the aid of a computer program in \texttt{Mathematica}.