AI 中文总结
研究舒尔数，提出移位S模板构造方法，通过增加着色灵活性及新的单元格着色方式，得到改进的递归式，结合已知边界，提升了S(8)和S(13)的下界。
AI 中文摘要
我们提出了一种由罗利开发的基于模板方法对舒尔数的扩展。这种新的模板构造形式，我们称为移位S模板，是在与ChatGPT 5.5 Pro的对话中发现的，然后经过完善、验证并扩展到多个移位S模板。这些新模板通过在着色上提供更多灵活性来推广最初的模板。利用这种增加的灵活性和给模板特殊标签单元格着色的新方法，我们展示了一个产生递归式S(k + 2)≥10S(k)+2的模板，改进了经典的阿博特 - 汉森递归式S(k + 2)≥9S(k)+4。结合已知边界S(6)≥536和S(11)≥203828，这意味着S(8)≥5362和S(13)≥2038282，改进了之前列出的下界5286和2011290。
英文摘要
We present an extension of the template-based approach for Schur numbers developed by Rowley. This new form of template construction, which we call shifted S-templates, was discovered during a conversation with ChatGPT 5.5 Pro, then refined, verified, and extended to multiple shifted S-templates. These new templates generalize the first ones by giving more flexibility in the coloring. Using this added flexibility with the new way to color the special label cells of the template, we exhibit a template which yields the recurrence $S(k+2) \geq 10S(k)+2$, improving on the classical Abbott-Hanson recurrence $S(k+2) \geq 9S(k)+4$ for the same step. Combined with the known bounds $S(6) \geq 536$ and $S(11) \geq 203\,828$, this implies $S(8) \geq 5\,362$ and $S(13) \geq 2\,038\,282$, improving the previously listed lower bounds $5\,286$ and $2\,011\,290$.
Comments4 pages, 4 figures, 2 tables
