无尽的平方自由数位走行
Dead ends in square-free digit walks
AI总结：
该研究证明了在平方自由数位走行中，死胡同的渐近密度远低于预测值，且通过形式化方法给出了闭合表达式。
摘要：
我们研究了平方自由数位走行中的'死胡同'：即平方自由整数N，使得在基数b下，每一个一位扩展bN+d都不是平方自由的。在十进制中，Miller等人提出的随机独立模型表明，无限平方自由走行出现的概率接近1，对应着渐近死胡同密度约为5.218×10^-5。我们证明了真正的渐近死胡同密度满足[c_dead ≈ 1.317×10^-9]，大约比预测值小一个数量级为4×10^4。对于每个基数b≥2，我们证明死胡同密度存在且由一个闭合表达式（作为有限交替欧拉乘积之和）给出。该论证在Lean/Mathlib中完全形式化，并由AxiomProver从问题的自然语言陈述自动生成。
英文摘要：
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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