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Walking to Infinity Along Some Number Theory sequences

Steven J. Miller, Fei Peng, Tudor Popescu, Joshua M. Siktar, Nawapan Wattanawanichkul, The Polymath REU Program

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.

Walking to Infinity Along Some Number Theory sequences

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.

Paper Structure

This paper contains 17 sections, 11 theorems, 60 equations, 9 tables, 4 algorithms.

Table of Contents

  1. Introduction
  2. Background
  3. Stochastic models
  4. Results
  5. Initial models of prime walks
  6. Models
  7. Results and comparison of models
  8. Main results for prime walks
  9. Modeling square-free walks
  10. Model
  11. Quantitative Results
  12. Additional remarks on the behavior of square-free walks
  13. Blind limited model
  14. Conclusion
  15. Impossibility of Walks
  16. ...and 2 more sections

Key Result

Theorem 2.3

Let $p_0$ be a prime. Then there exists a sequence of infinitely many primes $p_0, p_1, \ldots$ such that for all $i\ge1$, $p_i$ is equal to $10^{n_i}\cdot p_{i-1}+k_i$, for positive integers $n_i$ and $k_i$ with $k_i<10^{n_i}$.

Theorems & Definitions (32)

  • Theorem 2.3
  • proof
  • Definition 2.4
  • Lemma 2.5
  • proof
  • Theorem 2.6
  • proof
  • Theorem 2.7
  • proof
  • Definition 3.1: Square-Free Integer
  • ...and 22 more