Walking to Infinity on the Fibonacci Sequence

TL;DR

This paper investigates the possibility of constructing long sequences within the Fibonacci numbers by appending digits, proving that such walks are extremely limited in length and providing formulas for their maximum length under certain conditions.

Contribution

It introduces a novel analysis of digit-append walks on Fibonacci numbers, establishing upper bounds and formulas for their maximum lengths.

Findings

All walks starting with a Fibonacci number and appending one digit are at most length two.

Provides a formula for the maximum length of walks when appending a bounded number of digits.

Shows the rarity of long digit-append walks in Fibonacci numbers.

Abstract

An interesting open problem in number theory asks whether it is possible to walk to infinity on primes, where each term in the sequence has one more digit than the previous. In this paper, we study its variation where we walk on the Fibonacci sequence. We prove that all walks starting with a Fibonacci number and the following terms are Fibonacci numbers obtained by appending exactly one digit at a time to the right have a length of at most two. In the more general case where we append at most a bounded number of digits each time, we give a formula for the length of the longest walk.