How AI Is Uncovering New Ramanujan‑Style Formulas and Challenging Mathematics

Recent research published in *Nature* describes the Ramanujan Machine, an AI algorithm that automatically generates new mathematical formulas, many of which are difficult to prove; the project is named after the legendary Indian mathematician Srinivasa Ramanujan.

In April 2016, investor Yuri Milner hosted a small dinner at his home attended by tech leaders such as Google CEO Sundar Pichai, co‑founder Sergey Brin, Facebook CEO Mark Zuckerberg and dozens of other Silicon Valley figures. After screening a biographical film about Ramanujan, the guests announced a new fund to honor his legacy.

Ramanujan, one of the most iconic mathematicians of the 20th century, independently discovered nearly 3 900 formulas and propositions despite having little formal training; his intuition often produced correct theorems that were later proved, and his extensive notebooks continue to inspire research.

Some of his most celebrated formulas include:

The goal of the Ramanujan Machine is to devise new ways of computing important mathematical constants such as π or e, many of which are irrational and have infinite non‑repeating decimal expansions.

The algorithm starts from well‑known formulas (for example, the first few thousand digits of π), then attempts to predict a new expression that reproduces the same digits. The resulting conjecture must be validated by human mathematicians.

Since the project’s public launch in 2019, several of the generated conjectures have been proved correct, but challenges remain—for instance, the Apery constant, which has significant applications in physics, is still not fully understood.

Continued Fractions

Currently the algorithm can only produce a specific type of expression known as a continued fraction, which represents a number as an infinite nested fraction.

Researchers have applied this approach to various important constants, including Catalan’s constant (≈ 0.916), whose irrationality exponent was previously known to be at least 0.554. By using formulas generated by the Ramanujan Machine, the team improved this bound to 0.567.

Increasing Complexity

Automated conjecture generation is just one facet of computer‑assisted mathematics. Recent advances show that AI can not only perform repetitive calculations but also produce proofs, while software tools can verify human‑written proofs for correctness.

“Eventually humans may be replaced,” says mathematician Doron Zeilberger, a pioneer of proof automation, noting that as AI‑generated mathematics grows more complex, mathematicians may only be able to grasp the results at a high level.

Despite these advances, it remains unclear whether machines can distinguish deep, interesting statements from merely technically correct ones without human insight.

If you are interested, you can run the Ramanujan algorithm via the link below; any proved conjecture will be credited to you.

Reference: http://www.ramanujanmachine.com/

GitHub project: https://github.com/AnonGit90210/RamanujanMachine