Ramanujan Machine: GPU-Accelerated Formula Discovery

Abstract

We extend the Ramanujan Machine framework (Raayoni et al., 2024) using GPU-accelerated arbitrary precision arithmetic on an 8× B200 cluster. The original project evaluated ~1.77 million polynomial continued fractions (degree 2-3) over 2+ years of volunteer computing. Our goal: push to degree 4-6 polynomials and 10^9+ evaluations, potentially discovering new formulas for mathematical constants that the original search could not reach.

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	pi, ln(2), Gauss, Lemniscate
This work	2026	145 billion	1-5	None yet (all algebraic)

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 (2026-04-01)

Degree	Range	Candidates	Real Hits	Constants Found	Transcendental?
1	[-3,3]	2,401	~50	sqrt(2), phi	No
2	[-20,20]	4.75B	4.49M	sqrt(2), sqrt(5)	No
3	[-10,10]	37.8B	119M	sqrt(2)	No
4	[-5,5]	25.9B	260	sqrt(2)	No (2 false positive)
5	[-3,3]	13.8B	67.5K	sqrt(2)	No (2 false positive)
Total		~145B			Zero transcendental

Key observations

1. Degree 2 is fully exhausted at range [-20,20] (4.75B candidates). Only sqrt(2) and sqrt(5) matches.
2. Degree 4 produces dramatically fewer hits than degree 2-3 (260 vs millions) — the CF convergence is more selective at higher degree.
3. Two false positives at degree 4 matched pi·ln(2) at double precision but failed 50-digit mpmath verification. This confirms the need for PSLQ high-precision verification.
4. No formulas for pi, e, zeta(3), gamma, Catalan, or any transcendental constant through 145 billion candidates at degrees 1-5. Six false positives (pi/4, 2/sqrt(pi), pi·ln(2)) all failed 50-digit mpmath verification.

What this means

The Raayoni et al. results (degree 2-3, 1.77M candidates) found formulas for pi, e, ln(2), Gauss, and Lemniscate constants. Our search is 53,000× larger but at double precision instead of arbitrary precision. The lack of transcendental hits suggests either:

• Transcendental CF formulas are rare at these coefficient ranges and require larger coefficients or higher degrees
• Double-precision false-negative rate is high — real formulas may exist but converge too slowly for 500-term double-precision evaluation to distinguish them from noise
• Degree 5+ is where the action is — the polynomial structure may need to be richer

Next step: GPU PSLQ implementation for arbitrary-precision matching, and degree 5+ sweeps.

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.