In recent decades, a growing number of discoveries in fields of mathematics have been assisted by computer algorithms, primarily for exploring large parameter spaces that humans would take too long to investigate. As computers and algorithms become more powerful, an intriguing possibility arises - the interplay between human intuition and computer algorithms can lead to discoveries of novel mathematical concepts that would otherwise remain elusive. To realize this perspective, we have developed a massively parallel computer algorithm that discovers an unprecedented number of continued fraction formulas for fundamental mathematical constants. The sheer number of formulas discovered by the algorithm unveils a novel mathematical structure that we call the conservative matrix field. Such matrix fields (1) unify thousands of existing formulas, (2) generate infinitely many new formulas, and most importantly, (3) lead to unexpected relations between different mathematical constants, including multiple integer values of the Riemann zeta function. Conservative matrix fields also enable new mathematical proofs of irrationality. In particular, we can use them to generalize the celebrated proof by Ap\'ery for the irrationality of $\zeta(3)$. Utilizing thousands of personal computers worldwide, our computer-supported research strategy demonstrates the power of experimental mathematics, highlighting the prospects of large-scale computational approaches to tackle longstanding open problems and discover unexpected connections across diverse fields of science.

翻译：近几十年来，数学领域的发现越来越多地借助计算机算法辅助，主要用于探索人类难以在短时间内完成的巨大参数空间。随着计算机与算法日益强大，一个引人入胜的可能性浮现出来——人类直觉与计算机算法之间的相互作用可引导发现原本难以捕捉的新数学概念。为实现这一愿景，我们开发了一种大规模并行计算机算法，该算法发现了空前数量的基本数学常数连分数公式。算法所发现的巨量公式揭示了一种我们称之为保守矩阵场的新型数学结构。此类矩阵场：（1）统一了数千个现有公式；（2）生成无穷多个新公式；（3）更重要的是，建立了不同数学常数之间的意外关联，包括黎曼ζ函数在多个整数值处的关系。保守矩阵场还能实现无理性的新数学证明。特别地，我们可借助它们推广阿佩里关于ζ(3)无理性的著名证明。利用全球数千台个人计算机，我们的计算机辅助研究策略展现了实验数学的力量，凸显了大规模计算方法在解决长期未解难题及发现科学不同领域间意外关联方面的前景。