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Unsupervised Discovery of Formulas for Mathematical Constants
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Proposes dynamics-based metrics (irrationality measure via a Blind-delta algorithm, convergence-rate and denominator-growth fits) to cluster ~1.5M polynomial continued fractions, rediscovering known constant formulas and finding new ones for pi, ln(2), Gauss and Lemniscate constants.
Problem
Formulas for mathematical constants (continued fractions) lack a distance metric because a near-correct formula is simply wrong, so automated discovery and clustering of such formulas is hard.
Approach
Cluster polynomial continued fractions by their convergence dynamics (irrationality measure estimated without knowing the limit via the Blind-delta algorithm, plus fitted growth coefficients of error and denominators) rather than by coefficients or numerical value; use known formulas as anchors and PSLQ to label clusters.
Key results
- Irrationality measure delta achieves Davies-Bouldin index 1.11 versus 67.23 for the PCF limit L (lower is better).
- Clustering with 126 anchor formulas yields 441 novel automatically discovered formula hypotheses.
- Discovered novel formulas for pi, ln(2), Gauss and Lemniscate constants, with infinite families generalized from them; formal proofs left open.
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| Use irrationality measure δ instead of limit L as the clustering metric | PCF clustering · 25K converging PCFs | Davies-Bouldin Index: 67.23 -> 1.11 | Helped |
| Use denominator growth exponential coefficient γ′ instead of limit L as the clustering metric | PCF clustering · 25K converging PCFs | Davies-Bouldin Index: 67.23 -> 0.51 | Helped |
| Use denominator growth factorial coefficient η′ instead of limit L as the clustering metric | PCF clustering · 25K converging PCFs | Davies-Bouldin Index: 67.23 -> 0.13 | Helped |
| Use error rate exponential coefficient γ instead of limit L as the clustering metric | PCF clustering · 25K converging PCFs | Davies-Bouldin Index: 67.23 -> 14.83 | Helped |
| Use error rate factorial coefficient η instead of limit L as the clustering metric | PCF clustering · 25K converging PCFs | Davies-Bouldin Index: 67.23 -> 0.77 | Helped |
| Fit using 5 evenly spread points instead of 1000 points | PCF metric curve fitting · 450 randomly selected PCFs | Relative error average for denominator growth γ′: 0.0124 | No clear effect |
| Fit using 5 evenly spread points instead of 1000 points | PCF metric curve fitting · 450 randomly selected PCFs | Relative error average for denominator growth η′: 0.0007 | No clear effect |
| Fit using 5 evenly spread points instead of 1000 points | PCF metric curve fitting · 450 randomly selected PCFs | Relative error average for convergence rate γ: 0.0078 | No clear effect |
| Fit using 5 evenly spread points instead of 1000 points | PCF metric curve fitting · 450 randomly selected PCFs | Relative error average for convergence rate η: 0.0005 | No clear effect |
| Fit using 5 logarithmically spread points instead of 1000 points | PCF metric curve fitting · 450 randomly selected PCFs | Relative error average for denominator growth γ′: 0.0175 | No clear effect |
| Fit using 5 logarithmically spread points instead of 1000 points | PCF metric curve fitting · 450 randomly selected PCFs | Relative error average for denominator growth η′: 0.0004 | No clear effect |
| Fit using 5 logarithmically spread points instead of 1000 points | PCF metric curve fitting · 450 randomly selected PCFs | Relative error average for convergence rate γ: 0.0024 | No clear effect |
Surprising findings
- The PCF numerical limit is a very poor clustering feature, while dynamical metrics such as delta perform far better.
- Many novel conjectured formulas have a 'bad' irrationality measure near -1, which the authors argue is still informative under a non-judgmental metric.
Limitations
- Small set of metrics and a limited, low-degree PCF dataset (coefficients in [-5,5], degree up to 2).
- Novel formulas are numerically validated but not formally proven.
- Curve fits use only 5 points per metric, introducing approximation error.
Experimental setup
Datasets: polynomial continued fractions with coefficients in [-5,5] (1,768,900 formulas; 1,543,926 converging) 126 known anchor formulas for constants
Benchmarks: Davies-Bouldin Index on 25K sampled PCFs known formulas for pi, e, golden ratio, sqrt constants as anchors
Metrics: Davies-Bouldin Index irrationality measure PSLQ identification success
Baselines: PCF numerical limit clustering coefficient-based classification
Builds on: HDBSCAN Euler's continued fraction formula PSLQ
Closest papers by meaning
- From Euler to AI: Unifying Formulas for Mathematical Constants (NeurIPS 2025 · similarity 0.92)
- The Ramanujan Library - Automated Discovery on the Hypergraph of Integer Relations (ICLR 2025 · similarity 0.90)
- FIND: A Framework for Discovering Formulas in Data (AAAI 2025 · similarity 0.88)
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- Machine Learning meets Algebraic Combinatorics: A Suite of Datasets Capturing Research-level Conjecturing Ability in Pure Mathematics (ICML 2025 · similarity 0.84)
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