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Recently, Luca and Szalay solved the equation
Motivated by this equation, we show that the solutions of the equation
are (n, k, l, m) = (3, 2, 2, 2) and (5, 2, 1, 2). Here Fn is the nth Fibonacci number. To prove this, the main tools are linear forms in logarithm of algebraic numbers and the Lenstra-Lenstra-Lovász (LLL) lattice basis reduction algorithm.
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