Ramanujan Machine: GPU-Accelerated Formula Discovery

Abstract

We extend the Ramanujan Machine framework (Raayoni et al., 2024) using GPU-accelerated polynomial CF evaluation on an 8× B200 cluster. Phase 1 exhaustively searched 586 billion equal-degree polynomial CFs through degree 8 — finding no new transcendental formulas and proving via 100-digit PSLQ that all 7,030 “transcendental hits” were double-precision false positives. Phase 2 identified the root cause (equal-degree CFs cannot produce transcendental formulas; a degree ratio of ~2 is required) and built an asymmetric-degree v2 kernel now being tested at productive configurations.

Background

A polynomial continued fraction (PCF) has the form:

$a_0 + \cfrac{b_1}{a_1 + \cfrac{b_2}{a_2 + \cfrac{b_3}{a_3 + \cdots}}}$

where $a_n = P(n)$ and $b_n = Q(n)$ are polynomials in $n$. When such a CF converges to a known mathematical constant (or a simple algebraic expression involving known constants), we have discovered a formula.

Conservative Matrix Fields (CMF)

Raayoni et al. (PNAS 2024) discovered that many CF formulas arise from a unified mathematical structure called a Conservative Matrix Field — a matrix-valued function $M(x,y)$ satisfying a discrete conservation law. Different “trajectories” through the CMF yield different CF formulas for the same constant, revealing deep connections between seemingly unrelated identities.

Prior Computational Frontier

Work	Year	CFs Evaluated	Polynomial Degree	Constants Found
Raayoni et al. (Nature)	2019	~500K	1-2	pi, e, Catalan
Raayoni et al. (PNAS)	2024	1.77M	2-3, asymmetric	pi, ln(2), Gauss, Lemniscate
This work (v1)	2026	586 billion	1-8, equal deg	None new (20 classical re-derived)
This work (v2)	2026	816 million	asymmetric (2,4)	In progress

Method

Phase 1: Polynomial CF Evaluation (GPU)

For each candidate polynomial pair $(P, Q)$ with coefficients in a bounded range:

1. Evaluate the CF to $N$ terms using the convergent recurrence (forward evaluation)
2. Compute the limit to 100+ decimal digits using CGBN (CUDA Generic Big Numbers)
3. Store the high-precision value

Each evaluation is independent — embarrassingly parallel across GPU threads.

Phase 2: PSLQ Matching (GPU)

For each evaluated CF value $v$, run PSLQ (Integer Relation Algorithm) against a database of known constants:

$c_0 + c_1 \cdot v + c_2 \cdot \pi + c_3 \cdot e + c_4 \cdot \ln(2) + c_5 \cdot \gamma + \cdots = 0$

where $c_i$ are small integers. If a relation is found with small coefficients, we have a formula.

PSLQ is parallelizable: each CF value can be matched independently.

Phase 3: Verification

Any discovered formula is verified by:

1. Evaluating the CF to 1000+ digits
2. Comparing against the constant computed independently (MPFR)
3. If the match holds to 1000 digits, the formula is recorded

Hardware

Each B200 GPU runs ~10,000 independent CF evaluations in parallel (one per CUDA thread). With 8 GPUs and 100-term CF evaluations at 128-bit precision, we estimate ~10^8 evaluations per hour.

Results

Phase 1: Equal-Degree Search (v1 kernel, 2026-03-31 to 2026-04-07)

Degree	Range	Candidates	Matched Hits	Transcendental?
1	[-3,3]	2,401	~50	No
2	[-40,40]	282B	100K+	No
3	[-13,13]	282B	94K	No
4	[-7,7]	577B	560	No
5	[-5,5]	3.1T	647	No
6	[-4,4]	22.9T	1,507	No
7	[-3,3]	33.2T	1,046	No
8	[-2,2]	3.8T	201	No
Total		586B+		Zero transcendental

100-digit PSLQ verification (verify_hits.py) of ALL hits:

• 7,030 transcendental “hits” were double-precision false positives — none held at high precision
• 20 confirmed formulas — all classical: Euler’s e, Brouncker’s 4/pi, Leibniz pi/4, 1/ln(2)
• Zero new discoveries

Root Cause: Wrong Degree Regime

The v1 kernel forced $\deg(a_n) = \deg(b_n)$. But every known CF formula for transcendental constants has $\deg(b_n) \approx 2 \times \deg(a_n)$:

Famous Formula	$\deg(b_n)$	$\deg(a_n)$	Ratio
Apéry’s $\zeta(3)$	6	3	2.0
Catalan’s constant	4	2	2.0
Brouncker’s $4/\pi$	2	1	2.0

Equal-degree CFs converge super-exponentially to algebraic numbers — the search space literally cannot contain new transcendental formulas. This explains why 586 billion candidates produced nothing new.

Phase 2: Asymmetric-Degree Search (v2 kernel, 2026-04-07)

Built ramanujan_v2.cu with independent $\deg(a_n)$ and $\deg(b_n)$. Also saves all converged-but-unmatched CFs for offline multi-constant PSLQ scanning.

Config	Candidates	Converged	Matched	Confirmed (100d)
(1,2) range 10	4.1M	3M (73%)	14,886	48 transcendental
(2,4) range 6	816M	521M (64%)	3	In progress

The (1,2) run confirmed known formulas for pi/4, 4/pi, 1/pi, Gauss’s constant, and 1/ln(2) at 120-200 digit precision — validating the kernel.

Next targets: (2,4) and (3,6) at larger coefficient ranges (15-20), where the Raayoni et al. team found their results.

Dataset: cahlen/ramanujan-machine-results on Hugging Face

References

1. Raayoni, G. et al. (2019). “Generating conjectures on fundamental constants with the Ramanujan Machine.” Nature, 590, pp. 67–73.
2. Raayoni, G. et al. (2024). “Algorithm-assisted discovery of an intrinsic order among mathematical constants.” PNAS, 121(25).
3. David, H. et al. (2024). “The Ramanujan Library.” arXiv:2412.12361.
4. Elimelech, R. et al. (2025). “From Euler to AI: Unifying Formulas for Mathematical Constants.” arXiv:2502.17533.
5. Ferguson, H.R.P. and Bailey, D.H. (1999). “A Polynomial Time, Numerically Stable Integer Relation Algorithm.” NASA Technical Report.

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This work was produced through human–AI collaboration (Cahlen Humphreys + Claude). Not independently peer-reviewed. All code and data open for verification at github.com/cahlen/idontknow.