October 15, 2025

On k-generalized Fibonacci numbers which are perfect powers of Lucas numbers

Key Points

The research concludes that the equation $F_{n}^{(k)}=L_{m}^{a}$ has no solutions for $k\geq 3$, $m\geq 4$, and $a\geq 2$.
A critical component of the proof involves using lower bounds for linear forms in logarithms to establish the absence of solutions.
Understanding this relationship provides significant insights into the behavior of $k$-generalized Fibonacci numbers and Lucas numbers.
The methods applied also utilize tools from Diophantine approximation to strengthen the argument against possible solutions.

Abstract

Let (F₍^ (k) ) and (L₍) be the k-generalized Fibonacci and Lucas sequences, respectively. In this paper, we look at k-generalized Fibonacci numbers which are perfect powers of exponent larger than 1 of Lucas numbers. That is, we deal with the Diophantine equation F₍^ (k) =L₌^a in non-negative integers k, n, m, a, with k 3, m 4 and a 2. We show that this equation has no solution under these conditions. The proof depends on lower bounds for linear forms in logarithms and some tools from Diophantine approximation.

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