Epoch AI announced the first rational Diophantine septuple, a set of seven rational numbers where the product of any two plus one is a square, ending an 80-year conjecture by Erdős and Graham. The discovery, published February 2026, used an AI-guided search that pruned a candidate space of roughly 10^30 possibilities.
Key facts
- First rational Diophantine septuple found by AI
- Ends 1944 conjecture by Erdős and Graham
- Search space pruned by 99.99% from 10^30 candidates
- Verified with formal proof checker
- Published by Epoch AI, February 2026
Epoch AI announced the first rational Diophantine septuple, a set of seven rational numbers where the product of any two plus one is a perfect square, ending an 80-year conjecture by Erdős and Graham. The discovery, published February 2026, used an AI-guided search that pruned a candidate space of roughly 10^30 possibilities. According to Epoch AI, the result was verified by a formal proof checker, confirming the set satisfies the square condition for all 21 pairs.
The conjecture, raised by Paul Erdős and Ronald Graham in 1944, posited that no set of seven rational numbers could have this property. Prior work had found sextuples, but the jump to seven required a search space too large for brute force. The AI system, developed by Epoch AI's research team, combined number-theoretic constraints with neural guidance to prioritize promising branches, cutting the search time from an estimated centuries-long computation to days.
How the AI search cracked the problem
The algorithm didn't just enumerate candidates; it learned to avoid dead ends. By encoding Diophantine constraints into a loss function, the model generated partial sets and pruned those that violated the square condition early. This reduced the effective search space by 99.99%, according to the paper. The final septuple, which Epoch AI has not fully disclosed pending peer review, was checked with a formal proof assistant, closing the gap between heuristic search and mathematical certainty.
Critically, this isn't a brute-force win. The approach mirrors recent AI advances in mathematics, such as Google's DeepMind work on the cap set problem, but Epoch AI's method is distinct in its use of rational arithmetic, avoiding floating-point errors that would invalidate the proof. The result is a concrete example of AI not just conjecturing but delivering a verifiable theorem.
Why this matters beyond number theory
The discovery has implications for AI-driven mathematical research. Unlike prior AI results that were later found flawed, this one carries a formal proof, setting a standard for verifiability. It also suggests that AI can handle search problems where human intuition has stalled for decades, a pattern seen in Epoch AI's earlier work on FrontierMath benchmarks.
The septuple itself, while a curiosity, opens avenues in Diophantine geometry and elliptic curves. Researchers are already probing whether octuples exist, though Epoch AI notes the search space grows exponentially, making the next step computationally prohibitive without further algorithmic advances. The company did not disclose the compute budget, but sources suggest it used a cluster of H100 GPUs for under a week.
Key Takeaways
- Epoch AI used AI-guided search to find the first rational Diophantine septuple, ending an 80-year conjecture by Erdős and Graham.
- The result, verified by formal proof, shows AI can solve long-stalled number theory problems.
What to watch
Watch for the peer-reviewed publication of the septuple, expected within six months, and whether the method scales to octuples. Epoch AI's next benchmark release, likely late 2026, will show if the technique generalizes beyond Diophantine sets, with compute costs as the key metric.
Source: news.google.com
