cahlen/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).

Part of the 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 AI-audited, not peer-reviewed.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 (v2), ramanujan_gpu.cu (v1)
• Verification: verify_hits.py, pslq_scan.py
• Experiment: bigcompute.science

Citation

@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.