Ramanujan Machine

Ramanujan Machine on BOINC
Project
Status	Inactive
Category	Mathematics
Compute	CPU
Requires	None
Development
Developer	Technion – Israel Institute of Technology
Author	Ido Kaminer et al.
Sponsor	Technion – Israel Institute of Technology
Initial release	October 2021
Discontinued	2025
Software
Written in	C++
Operating system	Linux (AMD x86_64, Intel EM64T), Windows
BOINC statistics
Stats as of	23 May 2026
Active users	575
Total users	2,712
Active hosts	20,474
Total hosts	21,773
Metadata
Website	https://rnma.xyz/boinc/
License	See Source code

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Ramanujan Machine is a BOINC-based volunteer computing project that enlisted public computing power to discover new mathematical conjectures involving fundamental constants.

Background

The project takes its name from the self-taught Indian mathematician Srinivasa Ramanujan (1887–1920), who produced thousands of extraordinary mathematical formulas, many without formal proof.

The Ramanujan Machine software was conceptualized and developed at the Technion – Israel Institute of Technology under the guidance of Prof. Ido Kaminer, a member of the Andrew and Erna Viterbi Faculty of Electrical and Computer Engineering. The research began as an undergraduate project in the Rothschild Scholars Technion Program for Excellence, with participation from Gal Raayoni and George Pisha, and later expanded to involve Shahar Gottlieb, Yoav Harris, and Doron Haviv. A key algorithmic breakthrough was achieved by Shahar Gottlieb and led directly to the foundational paper's publication in Nature.

The details of the Ramanujan Machine were first published on 3 February 2021 in the journal Nature. The project was subsequently launched on BOINC in October 2021 in beta form, initially supporting only 64-bit Linux (AMD x86_64 and Intel EM64T architectures), with Windows support added later.

According to mathematician George Andrews, an expert on the mathematics of Ramanujan, while some results produced by the machine are remarkable and difficult to prove, they are not of the same caliber as Ramanujan's own work, and naming the software after him is "slightly outrageous." Conversely, Israeli mathematician Doron Zeilberger has described the Ramanujan Machine as a harbinger of a new methodology in mathematics.

Why Ramanujan Machine?

The Ramanujan Machine is a novel approach to mathematics: rather than relying on human intuition or ingenuity to discover formulas, it harnesses distributed computing power to search algorithmically for new mathematical relationships. Just as Ramanujan produced conjectures that he could not always prove, the machine generates conjectures without proving them, using AI and large-scale computation to "imitate" intuition.

As Prof. Kaminer has stated: "Our results are impressive because the computer does not care if proving the formula is easy or difficult, and does not base the new results on any prior mathematical knowledge, but only on the numbers in mathematical constants. To a large degree, our algorithms work in the same way as Ramanujan himself, who presented results without proof."

Goal

Fundamental constants like e and $\pi$ are ubiquitous in diverse fields of science, including physics, biology, chemistry, geometry, and abstract mathematics. Nevertheless, for centuries new mathematical formulas relating fundamental constants have been scarce and are usually discovered sporadically by mathematical intuition or ingenuity.

Ramanujan Machine algorithms search for new mathematical formulas expressed as polynomial continued fractions converging to fundamental constants. A general polynomial continued fraction takes the form:

$a_0+\frac{b_1}{a_1+\frac{b_2}{a_2+\frac{b_3}{a_3+\ddots }}}$

where $a_n$ and $b_n$ are polynomial functions of $n$. The project focused specifically on this class of representations because they offer elegant closed-form patterns even when capturing infinitely many terms, and because their convergence rate relates directly to the irrationality measure of the constant being approximated.

Methods

Algorithms

The Ramanujan Machine project has employed four major families of search algorithms:

• Meet-in-the-Middle (MITM-RF) – The original algorithm introduced in the 2021 Nature paper. It searches for continued fraction representations of fundamental constants by matching expansions computed from opposite ends of a search space.
• Gradient Descent (GD) – A numerical optimization approach that adjusts polynomial coefficients to drive convergence toward a target constant.
• Berlekamp–Massey – Identifies linear recurrences in integer sequences, enabling the discovery of formulas through pattern recognition rather than direct search.
• Factorial Reduction (LIReC) – A proprietary algorithm based on a phenomenon of factorial reduction in continued fraction denominators, described mathematically in Ben David et al. (2024). This algorithm is described as the most successful to date and forms the basis of the BOINC project's primary application, Ramanujan Machine New Key.

BOINC applications

The BOINC deployment ran three primary applications:

Application	Description
Ramanujan Machine New Key (rmach)	The main search application, based on the factorial-reduction irrationality measure phenomenon. Accounts for the vast majority of tasks distributed.
Blind Delta Algorithm (blind_delta_1)	A secondary search algorithm distributed to volunteers.
LIReC using BOINC instead of AWS	A workload for the Limitations of Irrationality measure using Continued fractions (LIReC) algorithm, formerly run on Amazon Web Services.

Conservative Matrix Fields

A major theoretical discovery arising from Ramanujan Machine research is the concept of the conservative matrix field (CMF). A CMF is a mathematical structure that unifies thousands of known polynomial continued fraction formulas into a single framework and generates infinitely many new ones from any given starting point. CMFs also reveal unexpected relationships among distinct mathematical constants, and have been used to re-motivate key steps in Apéry's proof that $\zeta (3)$ is irrational.

Project team

Srinivasa Ramanujan (1887–1920), the Indian mathematician after whom the project is named

The project was led by Prof. Ido Kaminer of the Viterbi Faculty of Electrical and Computer Engineering at the Technion. Key student contributors included:

• Gal Raayoni – lead author on the foundational Nature paper
• Shahar Gottlieb – developed the key breakthrough algorithm
• Yoav Harris
• Doron Haviv
• George Pisha
• Rotem Kalisch
• Rotem Elimelech
• Or David
• Carlos de la Cruz Mengual
• Mor Shalyt

Additional contributors are credited in the individual publications listed below.

Scientific results

The Ramanujan Machine has produced conjectures and proofs across several fundamental constants. For a full listing, see the official results page.

Conjectures for e

Early results included sets of continued fraction conjectures for Euler's number $e$, produced by the MITM-RF algorithm. An example of a proved result for $e$ is:

$e=2+\frac{1}{1+\frac{1}{2+\frac{2}{3+\frac{3}{4+\ddots }}}}$

Conjectures for $\pi$

Multiple sets of continued fraction conjectures were generated for $\pi$. A sample proved conjecture takes the form:

$\frac{\pi }{4}=1-\frac{1}{3+\frac{1\cdot 3}{5+\frac{2\cdot 4}{7+\frac{3\cdot 5}{9+\ddots }}}}$

Conjectures for ${\pi }^2$

About a dozen results involving ${\pi }^2$ were discovered, most remaining unproven as of the last project update.

Conjectures for $\log (2)$

A continued fraction for $\log (2)$ was discovered after May 2020 and is available on the project website.

Conjectures for Catalan's constant

In May 2020, the machine discovered several dozen conjectures for Catalan's constant $G$, most of which remain unproven.

Conjectures for Apéry's constant

Several conjectures were produced for Apéry's constant $\zeta (3)$. Two remain unproven as of the last update.

Riemann Zeta function conjectures

In June 2022, using a new algorithm, the machine discovered conjectures involving several different values of the Riemann zeta function. All of these remain unproven.

Ramanujan Library

A broader output of the project is the Ramanujan Library, introduced in 2024 at ICLR, which provides a hypergraph-based repository of integer relations among mathematical constants. The library automatically discovered 75 previously unknown connections among constants using the PSLQ algorithm.

Scientific publications

The following papers were produced by the Ramanujan Machine research group. The BOINC-specific algorithm is described in Ben David et al. (2024).

Primary paper

• (2021}).Generating conjectures on fundamental constants with the Ramanujan Machine. Nature. pp. 67–73. DOI: 10.1038/s41586-021-03229-4. – Introduces the MITM and gradient-descent algorithms; presents the first automated discovery of continued fraction conjectures for fundamental constants.

Further publications

• (2024}).On the Connection Between Irrationality Measures and Polynomial Continued Fractions. Arnold Mathematical Journal. DOI: 10.1007/s40598-024-00250-z. – Mathematical basis of the BOINC project's main algorithm; the most successful algorithm found in the project.

• (2024}).Algorithm-assisted Discovery of an Intrinsic Order Among Mathematical Constants. Proceedings of the National Academy of Sciences. pp. e2321440121. DOI: 10.1073/pnas.2321440121. – Introduces the conservative matrix field; unifies thousands of polynomial continued fraction formulas.

• Shalyt, M..(2024})."Unsupervised Discovery of Formulas for Mathematical Constants".

– Categorizes and identifies patterns in 1,768,900 unlabeled polynomial continued fraction formulas using convergence dynamics, rediscovering known formulas and uncovering new ones for $\pi$, $\ln (2)$, and other constants.

• Beit-Halachmi, I..(2025})."The Ramanujan Library – Automated Discovery on the Hypergraph of Integer Relations".

– Introduces the Ramanujan Library; discovers 75 previously unknown connections among mathematical constants.

• Razon, O..(2023})."Automated Search for Conjectures on Mathematical Constants using Analysis of Integer Sequences".pp. 28809–28842.

– Describes the ESMA algorithm for pattern search in integer sequences as an alternative conjecture-discovery method.

• David, O..(2023}).The conservative matrix field. arXiv. – Mathematical treatment of the CMF structure; provides a re-proof and motivation for Apéry's irrationality result for $\zeta (3)$.

• David, O..(2023}).On Euler polynomial continued fraction. arXiv. – Describes a method to identify and convert Euler polynomial continued fractions to infinite sums.

• Raz, T..(2025})."From Euler to AI: Unifying Formulas for Mathematical Constants".

– The Euler2AI project; uses LLMs and symbolic algorithms to scan hundreds of thousands of arXiv papers and find that 43% of $\pi$ formulas are unified within a single conservative matrix field. Featured in Scientific American for Pi Day 2026.

• Weinbaum, S..(2025})."On Conservative Matrix Fields: Continuous Asymptotics and Arithmetic".

– Connects CMFs to Ore algebras and D-finite functions; establishes striking numerical structure in irrationality measures.

• (2025}).ASyMOB: Algebraic Symbolic Mathematical Operations Benchmark. arXiv. – Introduces a 17,092-problem benchmark for evaluating LLMs on symbolic mathematics (integration, limits, differential equations); finds current LLMs rely heavily on pattern memorization and degrade up to 70% under slight problem variations.

External commentary

• Kures, Miroslav.(2022}).A note on the remarkable expression of the number $8/{\pi }^2$ that the Ramanujan Machine discovered. The Mathematical Intelligencer. pp. 150–152.

• Tonien, Joseph.(2026}).Franel Numbers and a Continued Fraction Conjecture Discovered by the Ramanujan Machine. The Mathematical Intelligencer. DOI: 10.1007/s00283-025-10497-9.

Server status

As of 23 May 2026, the server infrastructure at rnma.xyz is only partially operational. The download server, upload server, and scheduler are listed as running, but all background daemons — including the feeder, transitioner, file deleter, work generator, validators, and assimilators — are listed as not running. No tasks have been sent in recent 24 hours (0 hosts registered in past 24 hours). The transitioner backlog stands at over 5,252 hours, indicating the project has been in a non-functional state for many months.

Statistic	Value (as of 23 May 2026)
Users with credit	2,712
Users with recent credit	575
Hosts with credit	21,773
Hosts with recent credit	20,474
Current GigaFLOPS	11,251.16
Tasks ready to send	77,415
Tasks in progress	34,259
Transitioner backlog	5,252.73 hours

Source code

The Ramanujan Machine source code is hosted on GitHub at https://github.com/RamanujanMachine/. Repositories include the core algorithm library, the LIReC algorithm, and supplemental tools such as the Euler2AI and ASyMOB code bases.

See also

• Berkeley Open Infrastructure for Network Computing (BOINC)
• Srinivasa Ramanujan
• Continued fraction
• Apéry's constant
• Volunteer computing

External links

• Ramanujan Machine BOINC project
• Official Ramanujan Machine website
• Scientific results
• Publications
• GitHub repositories
• Project video (YouTube)