Fundamental mathematical constants appear in nearly every field of science, from physics to biology. Formulas that connect different constants often bring great insight by hinting at connections between previously disparate fields. Discoveries of such relations, however, have remained scarce events, relying on sporadic strokes of creativity by human mathematicians. Recent developments of algorithms for automated conjecture generation have accelerated the discovery of formulas for specific constants. Yet, the discovery of connections between constants has not been addressed. In this paper, we present the first library dedicated to mathematical constants and their interrelations. This library can serve as a central repository of knowledge for scientists from different areas, and as a collaborative platform for development of new algorithms. The library is based on a new representation that we propose for organizing the formulas of mathematical constants: a hypergraph, with each node representing a constant and each edge representing a formula. Using this representation, we propose and demonstrate a systematic approach for automatically enriching this library using PSLQ, an integer relation algorithm based on QR decomposition and lattice construction. During its development and testing, our strategy led to the discovery of 75 previously unknown connections between constants, including a new formula for the `first continued fraction' constant $C_1$, novel formulas for natural logarithms, and new formulas connecting $\pi$ and $e$. The latter formulas generalize a century-old relation between $\pi$ and $e$ by Ramanujan, which until now was considered a singular formula and is now found to be part of a broader mathematical structure. The code supporting this library is a public, open-source API that can serve researchers in experimental mathematics and other fields of science.

翻译：基础数学常数几乎出现在从物理学到生物学的每一个科学领域。连接不同常数的公式常常通过暗示先前不同领域之间的联系而带来深刻的洞见。然而，此类关系的发现仍然是稀少的事件，依赖于人类数学家零星的创造性突破。自动猜想生成算法的最新发展加速了对特定常数公式的发现。然而，常数之间联系的发现尚未得到解决。在本文中，我们提出了第一个专门致力于数学常数及其相互关系的库。该库可以作为来自不同领域科学家的核心知识库，以及作为新算法开发的协作平台。该库基于我们提出的用于组织数学常数公式的新表示形式：一个超图，其中每个节点代表一个常数，每条边代表一个公式。利用这种表示形式，我们提出并展示了一种使用PSLQ（一种基于QR分解和格构造的整数关系算法）自动丰富该库的系统性方法。在开发和测试过程中，我们的策略发现了75个先前未知的常数间联系，包括"第一连分数"常数 $C_1$ 的新公式、自然对数的新颖公式，以及连接 $\pi$ 和 $e$ 的新公式。后一类公式推广了拉马努金在一个世纪前发现的 $\pi$ 与 $e$ 之间的关系，该关系此前一直被视为一个孤立的公式，现在被发现是一个更广泛数学结构的一部分。支持该库的代码是一个公开的开源API，可以为实验数学和其他科学领域的研究人员提供服务。