Uniform Bounds for Digit-Appending Fibonacci Walks

TL;DR

This paper establishes uniform bounds on the ability to generate Fibonacci sequences through digit appending in various bases, demonstrating impossibility results that are independent of the base and extend to Lucas sequences.

Contribution

It provides a base-independent proof of bounds on Fibonacci walks with digit appending, extending results to Lucas sequences and improving understanding of sequence generation constraints.

Findings

Impossible to 'walk to infinity' with limited digit appending in any base

Derived a uniform bound L ≤ 2N log_φ b + O(1) for Fibonacci walks

Extended the approach to certain Lucas sequences

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.