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Dead ends in square-free digit walks

Evan Chen, Chris Cummins, Ben Eltschig, Dejan Grubisic, Leopold Haller, Letong Hong, Andranik Kurghinyan, Kenny Lau, Hugh Leather, Seewoo Lee, Aram Markosyan, Ken Ono, Manooshree Patel, Gaurang Pendharkar, Vedant Rathi, Alex Schneidman, Volker Seeker, Shubho Sengupta, Ishan Sinha, Jimmy Xin, Jujian Zhang

TL;DR

This work studies dead ends in base-$b$ square-free digit walks, defining a dead end as a square-free $N$ with every one-digit extension $bN+d$ non-square-free. It shows the stochastic prediction of dead-end density is drastically overestimated in base-10, proving $D(X)=c_{ ext{dead}}X+O(X/\sqrt{\log X})$ with $c_{ ext{dead}}\approx 1.317\times 10^{-9}$, far smaller than the predicted $\approx 5.218\times 10^{-5}$. The main method is a rigorous inclusion-exclusion sieve that expresses $c_{ ext{dead}}(b)$ as a finite alternating sum of Euler products, valid for all bases $b\ge 2$, and yields explicit numerical values for small prime bases. The results are fully formalized in Lean/Mathlib via AxiomProver, and the paper provides explicit computational details and a numerical strategy for $c_{ ext{dead}}$—highlighting arithmetic dependencies that suppress dead ends relative to naive stochastic models.

Abstract

We study "dead ends" in square-free digit walks: square-free integers $N$ such that, in base $b$, every one-digit extension $bN+d$ is non-square-free. In base $10$, the stochastic independence model of Miller et al. suggests that infinite square-free walks occur with probability near $1$, corresponding to an asymptotic dead-end density of $\approx 5.218\times 10^{-5}$. We prove that the true asymptotic dead-end density satisfies \[ c_{\mathrm{dead}} \approx 1.317\times 10^{-9}, \] roughly a factor of $\sim 4\times 10^4$ smaller than the prediction. For every base $b\geq 2$, we prove that dead-end densities exist and are given by a closed-form expression (as a finite alternating sum of Euler products). The argument is fully formalized in Lean/Mathlib, and was produced automatically by AxiomProver from a natural-language statement of the problem.

Dead ends in square-free digit walks

TL;DR

This work studies dead ends in base- square-free digit walks, defining a dead end as a square-free with every one-digit extension non-square-free. It shows the stochastic prediction of dead-end density is drastically overestimated in base-10, proving with , far smaller than the predicted . The main method is a rigorous inclusion-exclusion sieve that expresses as a finite alternating sum of Euler products, valid for all bases , and yields explicit numerical values for small prime bases. The results are fully formalized in Lean/Mathlib via AxiomProver, and the paper provides explicit computational details and a numerical strategy for —highlighting arithmetic dependencies that suppress dead ends relative to naive stochastic models.

Abstract

We study "dead ends" in square-free digit walks: square-free integers such that, in base , every one-digit extension is non-square-free. In base , the stochastic independence model of Miller et al. suggests that infinite square-free walks occur with probability near , corresponding to an asymptotic dead-end density of . We prove that the true asymptotic dead-end density satisfies roughly a factor of smaller than the prediction. For every base , we prove that dead-end densities exist and are given by a closed-form expression (as a finite alternating sum of Euler products). The argument is fully formalized in Lean/Mathlib, and was produced automatically by AxiomProver from a natural-language statement of the problem.
Paper Structure (10 sections, 12 theorems, 66 equations, 1 table)

This paper contains 10 sections, 12 theorems, 66 equations, 1 table.

Table of Contents

  1. Introduction
  2. Proof of Theorems \ref{['thm:main']} and \ref{['thm:base_b']}
  3. Inclusion-Exclusion Framework
  4. Asymptotics for $Q_S(X)$
  5. Proof of Theorem \ref{['thm:main']}
  6. Proof of Theorem \ref{['thm:base_b']}
  7. AxiomProver
  8. Appendix: Computing $\nu_p(S)$ and $c_{\mathrm{dead}}$
  9. Explicit computation of $\nu_p(S)$
  10. Computing $c_{\mathrm{dead}}$ numerically

Key Result

Theorem 1.1

For all $X\ge 3$, we have where the constant is and where for each subset $S\subseteq\mathcal{D}$ we define Moreover, have that $c_{\mathrm{dead}} \approx 1.3170\times 10^{-9}.$

Theorems & Definitions (27)

  • Conjecture : Miller et al. MillerEtAl
  • Remark
  • Definition
  • Theorem 1.1: Asymptotic density of dead ends
  • Remark
  • Theorem 1.2: Asymptotic density of dead ends in base $b$
  • Remark
  • Lemma 2.1: Chinese remainder theorem (CRT)
  • proof
  • Proposition 2.2: Exact digit inclusion--exclusion
  • ...and 17 more