Walking to Infinity Along Some Number Theory sequences

TL;DR

This paper investigates the possibility of infinite walks along primes and square-free numbers with increasing digit lengths, using greedy and stochastic models to analyze their long-term behavior and limitations.

Contribution

It introduces new greedy and stochastic models for prime and square-free walks, and proves the impossibility of infinite walks in certain number sequences and bases.

Findings

Greedy models predict long-term behavior of prime walks.

Stochastic models analyze expected walk length and digit frequency.

Proves impossibility of infinite walks in some sequences and bases.

Abstract

An interesting open conjecture asks whether it is possible to walk to infinity along primes, where each term in the sequence has one digit more than the previous. We present different greedy models for prime walks to predict the long-time behavior of the trajectories of orbits, one of which has similar behavior to the actual backtracking one. Furthermore, we study the same conjecture for square-free numbers, which is motivated by the fact that they have a strictly positive density, as opposed to primes. We introduce stochastic models and analyze the walks' expected length and frequency of digits added. Lastly, we prove that it is impossible to walk to infinity in other important number-theoretical sequences or on primes in different bases.