Terence Tao, the renowned mathematician and Fields Medal winner, recently highlighted a striking achievement by OpenAI’s GPT-5.2 Pro model: solving an open problem posed by Paul Erdős. In a post on X (formerly Twitter), Tao noted that the AI model successfully addressed a longstanding question in partition theory, but he cautioned that the accomplishment underscores computational speed rather than mathematical difficulty.
The problem in question originates from Erdős, one of the 20th century’s most prolific mathematicians, known for posing thousands of problems, many of which remain unsolved. This particular challenge involves partitioning the set of the first n positive integers, {1, 2, …, n}, into two subsets with sums as equal as possible. Erdős conjectured whether, for every sufficiently large n, such a partition always exists where the difference between the sums of the two subsets is at most 1. While the total sum of {1, 2, …, n} is n(n+1)/2, achieving near-perfect balance has proven nontrivial, especially for verifying the conjecture across large n.
GPT-5.2 Pro, an advanced iteration in OpenAI’s language model lineup, tackled this by performing exhaustive computational searches. The model confirmed the existence of such partitions for n up to extraordinarily large values, far beyond what manual computation or earlier algorithms could feasibly handle. Tao shared that the AI outputted explicit partitions for these cases, demonstrating not just existence but constructive solutions. This brute-force approach leveraged the model’s ability to run massive parallel computations internally, simulating searches that would take humans days or weeks.
However, Tao emphasized a critical nuance: the problem’s resolution by GPT-5.2 Pro reflects enhanced computational power more than profound insight. “This is more a win for speed than for difficulty,” Tao wrote. He explained that the issue is computationally intensive but lacks the deep structural barriers that characterize truly hard mathematical problems, such as those requiring novel theorems or paradigm shifts. Traditional methods, like dynamic programming or integer linear programming, scale poorly with n due to exponential time complexity. Humans might solve small instances by hand or with modest programs, but verifying the conjecture for n in the millions demands resources akin to supercomputing clusters.
Tao’s commentary aligns with ongoing debates about AI’s role in mathematics. Large language models like GPT-5.2 Pro excel at pattern recognition, optimization, and search tasks, often surpassing human limits in raw throughput. Yet, they rarely produce the elegant proofs or generalizations that define mathematical progress. In this case, the AI’s success stemmed from its training on vast datasets including computational techniques, allowing it to invoke optimized algorithms implicitly. Tao pointed out that while impressive, such feats do not equate to understanding; the model essentially accelerated known strategies rather than inventing new ones.
The broader context includes prior AI milestones in math. Systems like AlphaProof and Lean-assisted theorem provers have made headlines for formal proofs, but partition problems like Erdős’s lend themselves to verification over discovery. GPT-5.2 Pro’s performance here builds on its predecessors’ strengths in coding and simulation, where it can iterate through billions of possibilities in seconds. Tao’s post sparked discussions among mathematicians and AI researchers, with some praising the tool’s utility for conjecture-checking, others wary of overhyping its capabilities.
Erdős problems span number theory, combinatorics, and geometry, with bounties once attached to solutions. This instance, though not carrying a formal prize, exemplifies how AI lowers barriers to empirical validation. For researchers, tools like GPT-5.2 Pro now serve as high-speed oracles, testing hypotheses at scales previously inaccessible. Tao’s measured response tempers enthusiasm: while speed enables breakthroughs, true mathematical advancement demands creativity beyond computation.
This event also highlights evolving AI architectures. GPT-5.2 Pro incorporates reasoning chains and tool-use, enabling it to break down the partition task into subproblems: computing the total sum, targeting half-sum values, and backtracking for subsets. Its internal monologue, visible in outputs, reveals a step-by-step enumeration refined by beam search or Monte Carlo methods. Compared to earlier models, it handles memory-intensive states more efficiently, avoiding the combinatorial explosion that halts lesser systems.
As AI integrates deeper into research workflows, Tao’s observation serves as a reminder. Computational triumphs accelerate science, but distinguishing them from conceptual leaps remains essential. For the Erdős partition conjecture, GPT-5.2 Pro provides compelling evidence toward confirmation, pending formal proof. Mathematicians may now focus on generalization or related variants, armed with AI-generated data.
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