ramanujan-machine-results

# Ramanujan Machine — GPU Formula Discovery Results GPU-accelerated search for new continued fraction formulas for mathematical constants, inspired by [Raayoni et al. (2024)](https://www.pnas.org/doi/10.1073/pnas.2321440121). > Part of the [bigcompute.science](https://bigcompute.science) project. AI-audited, not peer-reviewed. ## Key Findings (Updated 2026-04-07) 1. **586 billion equal-degree polynomial CFs exhausted (v1 kernel, degrees 1-8)** — zero new transcendental formulas discovered. 2. **7,030 "transcendental hits" were double-precision false positives** — all disproven via 100-digit PSLQ verification (verify_hits.py). 3. **Only 20 confirmed formulas** — all classical: Euler's e, Brouncker's 4/pi, Leibniz pi/4, 1/ln(2). 4. **Root cause identified**: the v1 kernel forced deg(a_n) = deg(b_n), but every known CF formula for transcendental constants has **deg(b_n) ≈ 2 × deg(a_n)**. Equal-degree CFs converge super-exponentially to algebraic numbers and cannot produce new transcendental formulas. 5. **v2 kernel built** (ramanujan_v2.cu) with independent degrees for numerator and denominator polynomials. Validated on (1,2) regime — 48 confirmed transcendental formulas at 120-200 digit precision. ## v1 Results (Equal-Degree Search) | Degree | Range | Candidates | Constants Found | |--------|-------|-----------|-----------------| | 1-3 | up to [-40,40] | ~282B | sqrt(2), sqrt(5), phi only | | 4 | [-7,7] | 577B | sqrt(2) only | | 5 | [-5,5] | 3.1T | sqrt(2) only | | 6-8 | [-2,2] to [-4,4] | ~60T | sqrt(2) only | | **Total** | | **586B+** | **Zero new transcendental** | ### PSLQ Verification (2026-04-07) - 7,030 claimed transcendental matches → **all false positives** at 100-digit precision - 20 confirmed formulas → **all classical, previously known** - Additional tests: deeper CF evaluation (depth 5000), expanded constant library (30 constants incl. MZVs, Glaisher, Khinchin), rational coefficients — all negative ## v2 Results (Asymmetric-Degree Search, In Progress) | Config (deg_a, deg_b) | Range | Candidates | Converged | Confirmed | |---|---|---|---|---| | (1, 2) | [-10,10] | 4.1M | 3M (73%) | 48 transcendental (classical) | | (2, 4) | [-6,6] | 816M | 521M (64%) | In progress | The v2 kernel also saves all converged-but-unmatched CFs to enable offline multi-constant PSLQ scanning (pslq_scan.py). ## Method ### v1 (deprecated) For polynomial pairs (P, Q) of the same degree with bounded integer coefficients, evaluate the generalized CF to double precision (500 terms), then match against 10 base constants + 29 compound expressions. ### v2 (current) CF = a(0) + b(1) / (a(1) + b(2) / (a(2) + ...)) where a(n) has degree d_a and b(n) has degree d_b independently. Target: d_b ≈ 2 × d_a (the "productive zone" for transcendental constants). GPU evaluates at double precision; survivors verified via CPU PSLQ at 100+ digits. ## Understanding This Data The Ramanujan Machine project tries to discover new mathematical formulas by brute force: generate billions of continued fraction expressions and check whether any of them equal known constants like pi, e, or zeta(3). **The key lesson from this dataset**: the polynomial degree structure matters more than the search volume. 586 billion equal-degree candidates produced nothing, while 4 million asymmetric-degree candidates immediately re-derived classical formulas. The productive zone is deg(numerator) ≈ 2 × deg(denominator), matching the theoretical insight from Raayoni et al.'s Conservative Matrix Field framework. ## Source - **Code**: [ramanujan_v2.cu](https://github.com/cahlen/idontknow/blob/main/scripts/experiments/ramanujan-machine/ramanujan_v2.cu) (v2), [ramanujan_gpu.cu](https://github.com/cahlen/idontknow/blob/main/scripts/experiments/ramanujan-machine/ramanujan_gpu.cu) (v1) - **Verification**: [verify_hits.py](https://github.com/cahlen/idontknow/blob/main/scripts/experiments/ramanujan-machine/verify_hits.py), [pslq_scan.py](https://github.com/cahlen/idontknow/blob/main/scripts/experiments/ramanujan-machine/pslq_scan.py) - **Experiment**: [bigcompute.science](https://bigcompute.science/experiments/ramanujan-machine-gpu/) ## Citation ```bibtex @misc{humphreys2026ramanujan, author = {Humphreys, Cahlen and Claude (Anthropic)}, title = {Ramanujan Machine: GPU-Accelerated CF Formula Discovery}, year = {2026}, publisher = {Hugging Face}, url = {https://huggingface.co/datasets/cahlen/ramanujan-machine-results} } ``` Human-AI collaborative work. AI-audited against published literature. Not independently peer-reviewed. CC BY 4.0.