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arXiv · 2602.05095

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
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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.

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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. 2026-02-04. Dead ends in square-free digit walks. https://arxiv.org/abs/2602.05095

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