Core Insights - The article discusses the successful proof of an Erdős conjecture by OpenAI's latest model, GPT-5.2 Pro, marking a significant achievement in the intersection of AI and mathematics [2][3]. - The proof was validated by Fields Medalist Terence Tao, who described it as "the clearest first-class result contributed by AI to date" [3]. Group 1: The Proof and Its Validation - The conjecture, known as problem number 281 in the Erdős problem collection, was proposed in 1980 by mathematicians Paul Erdős and Ronald Graham, relating to deep connections between congruence covering systems and natural density [4][5]. - The proof utilized the infinite Adèle ring, employing Haar measure and pointwise ergodic theorems, transitioning from pointwise convergence to uniform convergence [9][10]. - Tao confirmed the proof's validity by translating the ergodic argument into combinatorial language, using the Hardy-Littlewood maximal inequality instead of the Birkhoff theorem [16]. Group 2: Alternative Solutions and Historical Context - An unexpected discovery emerged when a user named KoishiChan pointed out that a simpler solution to the problem exists, utilizing two theorems established in 1936 and 1966 [18]. - The first theorem is the density convergence theorem co-proven by Harold Davenport and Erdős in 1936, and the second is Rogers' theorem, first published in 1966 [19]. - Tao noted that Erdős himself was unaware of this simpler solution when he proposed the problem in 1980, raising questions about the problem's formulation [20]. Group 3: AI's Performance and Future Implications - Following the announcement, various AI models were tested for cross-validation, with Gemini 3 Pro confirming the proof's correctness [24]. - However, Tao cautioned about the statistical bias in evaluating AI's success rates, highlighting that negative results are often underreported [27]. - Current data suggests that the real success rate of these tools on Erdős problems is approximately 1% to 2%, which still indicates a significant number of non-trivial contributions from AI [31][32].

Core Viewpoint - OpenAI's latest model, GPT-5.2 Pro, has independently proven a conjecture from the Erdős problem set, marking a significant achievement in AI's capability in mathematical reasoning [1][2]. Group 1: The Proof and Its Validation - The conjecture, known as the 281st problem in the Erdős problem library, was proposed in 1980 and remained unsolved for 45 years until a researcher named Neel Somani presented it to GPT-5.2 Pro [2]. - The proof utilized the infinite Adele integer ring and involved the Haar measure and pointwise ergodic theorem, transitioning from pointwise convergence to uniform convergence [3]. - The proof has been validated by Fields Medalist Terence Tao, who noted that GPT-5.2 Pro did not make common errors typically seen in previous AI models, such as mistakes in limit exchanges or quantifier order [8]. Group 2: Alternative Solutions and Historical Context - An unexpected discovery revealed that the problem has a simpler solution using two theorems established in 1936 and 1966, namely the density convergence theorem and Rogers' theorem [9]. - Terence Tao communicated with mathematician Tenenbaum, who confirmed that the problem could be solved using these classical results, suggesting that the problem's statement might have been altered at some point [10]. Group 3: AI's Performance and Statistical Insights - Following the announcement, various AI models were tested for cross-validation, with Gemini 3 Pro confirming the proof's validity [11]. - However, Tao cautioned that the true success rate of AI in solving mathematical problems is likely skewed due to reporting bias, as negative results are less likely to be shared [11]. - Current data indicates that AI tools have a real success rate of approximately 1% to 2% on Erdős problems, which, despite being low, still represents a significant number of non-trivial contributions given the existence of over 600 unsolved problems in the library [12].