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.