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arXiv · 2603.21259

On mixed $b$-concatenations of Fibonacci and Lucas numbers that are Lucas numbers

Herbert Batte·Prosper Kaggwa
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Abstract

Let $(F_n)_{n\ge0}$ and $(L_n)_{n\ge0}$ denote the sequences of Fibonacci and Lucas numbers respectively. This paper determines all Lucas numbers that can be represented as base $b$ mixed concatenations of a Fibonacci number and a Lucas number. Mathematically, we study of two Diophantine equations $L_n=b^dL_m+F_k$ and $L_n=b^dF_m+L_k$, where $d$ is the number of digits of $F_k$ or $L_k$ in base $b$. To tackle these equations, we combine tools from Diophantine approximation on non-zero linear forms in logarithms and reduction methods based on continued fractions. This allows us to prove that only finitely many such Lucas numbers exist.

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Herbert Batte, Prosper Kaggwa. 2026-03-22. On mixed $b$-concatenations of Fibonacci and Lucas numbers that are Lucas numbers. https://arxiv.org/abs/2603.21259

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