In an unprecedented scientific achievement reflecting the rapid evolution of artificial intelligence technologies, recent reports indicate that OpenAI’s “GPT-5.4 Pro” model has successfully cracked the open Erdős problem No. 1196. This problem is considered a complex mathematical conjecture regarding primitive sets of integers, which has resisted solution for nearly 60 years. Surprisingly, the model managed to find the final solution in approximately 80 minutes, and did not stop there; it took another 30 minutes to prepare and format the solution into a complete academic research paper using a specialized mathematical typesetting system. This achievement has been widely discussed across the Erdős Problems project forum, where formal step-by-step verification of the mathematical proof is currently underway.
A previously undescribed scientific link
The roots of this complex problem date back to the 1960s, when it was posed by legendary Hungarian mathematician Paul Erdős in collaboration with fellow researchers Sárközy and Szemerédi. The dilemma focuses on the behavior of a specific sum over primitive sets, which are sets of integers characterized by unique properties where no number is divisible by another within the same set. Over past decades, many have attempted to find a definitive solution, most notably mathematician Jared Duker Lichtman, who succeeded in establishing an upper bound of approximately 1.399 for the sum in question, yet the full asymptotic conjecture remained an open mystery waiting to be decoded.
The major breakthrough came when Epoch AI researcher Liam Price prompted the model to work on the problem. Here, the advanced AI model relied on a technique known as Markov chains, cleverly combining it with von Mangoldt weights. This integration represents an innovative approach that no human mathematician had attempted to use, despite years of prolonged effort spent trying to solve this dilemma.
In this context, Terence Tao, a mathematician awarded the prestigious Fields Medal, commented via the Erdős Problems forum, noting that this work reveals a deep link between the structure of integers and the theory of Markov processes—a link never before described in mathematical literature. Tao emphasized that this discovery is a highly valuable contribution to understanding the anatomy of integers, and its importance transcends merely solving this specific problem, adding that the core step in the proof involved a genuine qualitative leap, even if it seemed obvious in hindsight.
Meanwhile, Kevin Barreto, a 21-year-old undergraduate mathematics student at the University of Cambridge who announced he will soon join OpenAI’s AI for Science team, noted that using the Markov chain method represents an exceptional creative step absent from previous human attempts. This confirms the machine’s ability to think outside the box and provide completely fresh perspectives on classical problems.
The accelerating role of AI in mathematics
This achievement is not an isolated event, but part of a broader wave of mathematical breakthroughs recently driven by intelligence. The same model successfully delivered verified solutions to Erdős problems No. 1148 and No. 1202, and also managed to solve a previously unsolved problem concerning hyper-Ramsey graphs, within the mathematical evaluation benchmark established by Epoch AI.
The pace of this stunning progress has accelerated since last summer, when both Google DeepMind and OpenAI achieved gold-medal-equivalent results at the 2025 International Mathematical Olympiad held in Australia, with each model solving 5 out of 6 highly difficult problems included in the competition. This raises serious questions about the future of scientific research in mathematics and the role of machines within it.
Do language models invent new knowledge?
With these successive achievements, the philosophical and scientific debate continues over whether large language models are truly capable of generating entirely new mathematical knowledge, or if they merely recombine and synthesize information learned during training. As recent analytical reports noted, the stunning solution to the Erdős problem provides strong evidence that new knowledge not previously described can be hidden within already familiar data points, and what it requires is extraordinary computational and analytical power to connect those dots and extract innovative solutions from them.
Frequently Asked Questions
What is Erdős problem No. 1196?
It is a mathematical problem posed by Paul Erdős in the 1960s regarding the behavior of primitive sets of integers, where no number in these sets is divisible by any other number within them, and the problem remained completely unsolved for nearly 60 years.
How long did it take the model to reach the solution?
The model took about 80 minutes to arrive at the mathematical solution, in addition to another 30 minutes to write and format the solution into a complete scientific research paper.
What is the new approach used by the model?
The model used a technique known as Markov chains combined with von Mangoldt weights, an inventive method that human mathematicians had never used in their attempts to solve this dilemma.
What is the significance of this achievement for the future of artificial intelligence?
This achievement proves that artificial intelligence models are no longer content with recycling data, but have become capable of inventing new methods and discovering previously unknown scientific links, opening wide horizons in the field of scientific and mathematical research.
