The Ramanujan Library -- Automated Discovery on the Hypergraph of Integer Relations

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 $π$ and $e$. The latter formulas generalize a century-old relation between $π$ 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.

💡 Research Summary

The paper introduces the Ramanujan Library, the first publicly available repository that systematically collects mathematical constants and the formulas that relate them, organized as a hypergraph. In this hypergraph each vertex represents a constant (either a well‑known constant such as π, e, ζ(2) or a constant defined by a convergent continued‑fraction or infinite sum) and each hyperedge encodes an integer relation among a set of constants. The authors adopt a canonical C‑transform notation to express continued fractions and infinite series uniformly, allowing any formula to be treated as a formal expression that can be evaluated to high precision when it converges.

The core discovery engine is the PSLQ integer‑relation algorithm, which uses QR decomposition and lattice reduction to find integer vectors a₁,…,aₙ such that Σ aᵢxᵢ≈0 for a given high‑precision vector (x₁,…,xₙ). Because PSLQ can produce spurious relations when the working precision is limited, the authors devise a new heuristic called Return on Investment (RoI). RoI combines the binary length of the discovered integers with the overall precision of the computation; empirically they find that RoI≥2 reliably separates true relations from random artefacts.

The enrichment process proceeds in two stages. First, a database of fundamental constants and C‑transform‑generated constants is built. The user defines subsets of constants and a maximum polynomial degree d and order o. All Cartesian products of the chosen subsets are generated, and for each product PSLQ is run. When a relation passes the RoI threshold it is added to the hypergraph as a hyperedge, and the precision of the relation is recorded as ⌊−log₂ ε⌋ where ε is the residual error. Second, the hypergraph is used to prune future searches: if a candidate set already contains an edge of degree ≤d and order ≤o, PSLQ is skipped, dramatically reducing computational cost. This “learning” aspect makes the algorithm embarrassingly parallel and scalable across many processors.

Applying this pipeline, the authors discovered 75 previously unknown integer relations. Highlights include a new formula for the “first continued‑fraction constant” C₁, several novel identities involving natural logarithms, and a family of polynomial relations linking π and e. The latter generalize the classic Ramanujan identity π·e = 1+1/(1+2/(1+3/…)), showing that π and e participate in an entire hierarchy of nonlinear relations rather than a single isolated formula.

All code, data, and the API are released under an open‑source license, enabling researchers to add new constants, query the hypergraph, or run custom PSLQ searches. The API supports batch submission of constant sets and returns discovered relations together with their RoI scores, facilitating integration with other experimental‑mathematics platforms.

The significance of the work lies in (1) providing a formal graph‑theoretic framework that captures both linear and nonlinear connections among constants, and (2) demonstrating that automated, high‑precision integer‑relation searches can systematically expand this framework beyond what human intuition alone can achieve. The authors suggest future extensions such as exploring higher degree and order bounds, incorporating alternative integer‑relation algorithms like LLL, and linking the library with empirical data from physics, biology, or engineering to uncover interdisciplinary constant relationships. The Ramanujan Library thus represents a concrete step toward a shared, machine‑readable knowledge base for fundamental constants, opening new avenues for both pure number theory and applied scientific discovery.