Research map / Probabilistic and scientific modeling / Molecules and proteins
The Ramanujan Library - Automated Discovery on the Hypergraph of Integer Relations
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Builds the Ramanujan Library, a hypergraph of mathematical constants and their integer polynomial relations, and automatically enriches it with PSLQ plus a Return-on-Investment filter, yielding 75 new relations.
Problem
Connections between fundamental mathematical constants (e.g., pi, e, ln2, zeta values) are discovered only sporadically by human mathematicians; no systematic library or automated method exists for discovering relations between constants.
Approach
Represent constants and formulas (C-transforms of continued fractions) as a hypergraph where vertices are constants and hyperedges are integer polynomial relations, and populate it automatically by running PSLQ over constant subsets, filtering results with a novel Return-on-Investment (RoI) significance measure.
Key results
- Automated search yielded 118 relations between constants, of which 75 are novel and 43 known in the literature.
- The RoI cutoff of 2 was proposed to separate significant PSLQ relations from random false positives.
- identify succeeded on C-transform cases where Wolfram Alpha failed, e.g., finding the limit (5A+6B)/(5A-9B) in terms of Lemniscate constants.
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| Use the automated PSLQ-based search to discover relations between constants | Ramanujan library · Mathematical constants and continued fractions | novel connections discovered: 75 | Helped |
| Use polynomial integer relations rather than restricting the search to linear relations | Automated relation search · Mathematical constants and continued fractions | relation types discovered | Helped |
| Use identify for numerical identification instead of Wolfram Alpha | Numerical identification of continued-fraction limits · Lemniscate-constant formula | constant identification success | Helped |
Surprising findings
- Ramanujan's pi-e relation generalizes to a family of 8 formulas for sqrt(pi e), so it is not a singular formula.
- PSLQ runs can exhibit a rebounding phenomenon where precision first rises then falls, requiring a failsafe.
Limitations
- Results are numerical conjectures, not proofs; only 47 formulas were proven.
- The search relies on a pre-selected static set of constants.
Experimental setup
Datasets: database of fundamental constants and continued fractions from the literature
Benchmarks: Wolfram Alpha numerical identification comparison
Metrics: precision (binary digits) return on investment (RoI) number of novel relations found
Baselines: Wolfram Alpha
Builds on: PSLQ LLL Ramanujan Machine Euler continued fraction transform
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