OpenAI Solves Three Erdős Problems: Internal Model Outperforms Public Counterparts
OpenAI's internal model independently proved three open problems posed by Paul Erdős decades ago. The preprint authors—Boris Alekseev, Mo Patterson, Mehtab Soni, Mark Sellke, and Gregory Valiant—confirmed that the proofs were generated entirely by AI, with their role limited to editing for clarity. The public GPT-5.4 Pro managed only two out of three problems, highlighting the superiority of the internal system.
Solution to Problem #684: Polylogarithmic Bound
Problem #684 from Erdős's list concerns small prime factors of binomial coefficients. The model established a polylogarithmic upper bound on their frequency. Previous results were limited to subpolynomial bounds, making this progress significant.
This achievement demonstrates AI's ability to work with asymptotic estimates in combinatorics. The proof relies on probabilistic methods and properties of primes in binomial coefficients, improving theoretical bounds for further research.
Counterexample for Problem #741
Problem #741 poses a question by Burr and Erdős about partitioning an additive basis of order two into two parts. The sums in each part must have bounded gaps. The model constructed an explicit counterexample, disproving the conjecture.
Erdős considered the problem solvable but never completed the proof. The counterexample uses a construction of sets with controlled densities, violating the bounded gaps condition in the sums. This is the first rigorous disproof of the conjecture.
- Key elements of the counterexample:
- Additive basis of order two with specified density.
- Partition into subsets with asymmetric sum properties.
- Demonstration of unbounded gaps in one of the parts.
Full Proof of Hypothesis #997
Problem #997 (1964) states that the fractional parts {α p_n} (where p_n are primes) are never well-distributed. In 2024, Champagne, Le, Liu, and Wu proved this for a specific α. The OpenAI model generalized the result to all α.
The proof combines analytic number theory and properties of prime sequences. It shows the lack of uniformity in the distribution via a variance estimate.
Comparison with Public Models
Testing on GPT-5.4 Pro revealed limitations:
- Problem #684: solved in <10 attempts.
- Problem #741: unsolved even after multiple attempts.
- Problem #997: solved in <10 attempts.
Failure on #741 points to the higher capabilities of the internal model, presumably Spud. This confirms gaps in public versions.
Historical context:
- October 2025: GPT-5 'solved' 10 problems but copied from literature.
- January 2026: GPT-5.2 Pro + Aristotle solved #728—the first original AI proof.
The current preprint sets a new standard: three original proofs without human mathematical input.
What Matters
- OpenAI's internal model generates original proofs of Erdős problems unavailable to public versions.
- Polylogarithmic bound for #684 improves prior subpolynomial bounds.
- Counterexample for #741 disproves the Burr–Erdős conjecture on bases.
- Full proof of #997 for all α closes a 60-year-old problem.
- Tests on GPT-5.4 Pro highlight the capability gap between models.
— Editorial Team
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