Lemma Park

Problems we're exploring

1. Open Erdős problems in Lean

   Paul Erdős, one of the most prolific mathematicians of the last century, left hundreds of problems that are easy to state and hard to solve, many with small cash prizes attached. Most are still open. They have become a common test of whether AI can do new mathematics rather than repeat what it has read.

2. Primes in progressions, Euclid's way

   Euclid proved over two thousand years ago that the primes never run out. A harder question is whether primes of a given kind never run out, such as primes that leave remainder 1 when divided by 8. Euclid's simple style of proof answers it for some kinds and not others, and this program tests how many of those cases an AI can prove, with every step checked by a computer.

3. Classic theorems without their usual tools

   Famous theorems are usually proved with powerful general tools, and it is hard to tell how much of that power they really need. Here agents prove seven classic theorems with the usual tools taken away, as a biologist learns what a gene does by switching it off. Terence Tao has suggested this kind of study as a good use of AI in mathematics.

4. Fast 5×5 matrix multiplication

   Computers spend much of their time multiplying tables of numbers, called matrices: it is most of the work in running AI models and in simulating weather or physics. Fewer multiplications means the same answer in less time and with less power, and recipes for small tables are the building blocks for large ones. In 2025 computer searches cut the best known recipe for 5×5 from 97 multiplications to 93, and nobody knows how low it can go.

   Our best: 93

   Record: 93

   Lower is better

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