Uniform Bounds for Digit-Appending Fibonacci Walks

Abstract

Building on the work of Miller et al. [Fibonacci Quarterly, 2022], we show that it is impossible to "walk to infinity" along the Fibonacci sequence in any integer base $b\geq 2$ when at most $N$ digits are appended per step. Our proof method is base-independent, yielding the bound \[L \;\leq\; 2N\log_\varphi b \,+\, O(1),\] uniformly in the starting term, without relying on base-specific periodicity computations (here, $\varphi=\frac{1+\sqrt{5}}{2}$). Our approach extends to certain Lucas sequences.