In 1989, computer searches by Lam, Thiel, and Swiercz experimentally resolved Lam's problem from projective geometry$\unicode{x2014}$the long-standing problem of determining if a projective plane of order ten exists. Both the original search and an independent verification in 2011 discovered no such projective plane. However, these searches were each performed using highly specialized custom-written code and did not produce nonexistence certificates. In this paper, we resolve Lam's problem by translating the problem into Boolean logic and use satisfiability (SAT) solvers to produce nonexistence certificates that can be verified by a third party. Our work uncovered consistency issues in both previous searches$\unicode{x2014}$highlighting the difficulty of relying on special-purpose search code for nonexistence results.

中文翻译：

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基于SAT的林氏问题解决

1989年，Lam，Thiel和Swiercz进行的计算机搜索从射影几何学\ unicode {x2014}开始实验性地解决了Lam的问题，这是确定是否存在10阶射影平面的长期存在的问题。最初的搜索和2011年的独立验证都没有发现这种射影飞机。但是，这些搜索都是使用高度专业的自定义编写的代码执行的，并且不会生成不存在的证书。在本文中，我们通过将问题转化为布尔逻辑来解决Lam问题，并使用可满足性（SAT）求解器来生成不存在的证书，该证书可以由第三方进行验证。我们的工作在先前的搜索\\ unicode {x2014} $中都发现了一致性问题，突出显示了依靠专用搜索代码获得不存在结果的难度。