By Joe Udo
In a stunning demonstration of artificial intelligence’s growing prowess in pure mathematics, OpenAI’s ChatGPT-5.4 Pro model has solved Erdős open problem, for any x, if A⊂[x,∞ ), a conjecture dating back to the 1960s, in just 80 minutes, producing a LaTeX-formatted proof paper in another 30.
The breakthrough, announced on the Erdős Problems Project forum, tackles a question posed by legendary mathematician Paul Erdős alongside Sárközy and Szemerédi. It concerns the asymptotic behaviour of a weighted sum over “primitive sets” of integers, where no element divides another. Human efforts had narrowed the bound to roughly 1.399, but the full conjecture eluded researchers for decades.
Prompted by Epoch AI researcher Liam Price, the model introduced a Markov chain approach combined with von Mangoldt weights, a technique overlooked by human mathematicians. Fields Medalist Terence Tao praised the result on the forum, highlighting a “previously undescribed connection” between integer anatomy and Markov process theory.
“That would be a meaningful contribution… that goes well beyond the solution of this particular Erdős problem,” Tao wrote, noting the “leap” in the key step that now seems obvious in hindsight.
Cambridge math undergraduate Kevin Barreto, soon joining OpenAI’s AI for Science team, called it a creative leap prior efforts missed. Formal verification of the proof is underway.
This marks the third Erdős problem cracked by GPT-5.4 Pro, following #1148 and #1202, plus a Ramsey hypergraph solution from Epoch AI’s FrontierMath benchmark. It builds on AI’s gold-medal wins at the 2025 International Mathematical Olympiad by OpenAI and Google DeepMind.
A conjecture of Erdős, Sárközy, and Szemerédi. Lichtman [Li23] has proved that:
∑a∈A1aloga<eγπ4+o(1)≈1.399+o(1).This was solved by GPT-5.4 Pro (prompted by Price), which proved that for any primitive set A⊂N
∑a∈Aa>x1aloga≤1+O(1logx).See the comment section for further refinements and discussion.
Lichtman [Li20] proved that if A is the set of all integers with exactly k prime factors (so that A⊂[2k,∞) and A is a primitive set) then
∑a∈A1aloga≥1+O(k−1/2+o(1)),and suggested that the true rate of decay may be O(2−k). Gorodetsky, Lichtman, and Wong [GLW24] have proved that
∑a∈A1aloga=1−(c+o(1))k22−kwhere c≈0.0656 is an explicit constant.
x=1.
The feat reignites debate: Can LLMs generate truly novel math, or merely recombine training data? As The Decoder observed, this suggests “new knowledge can also be hidden within already known data points.”
Mathematical communities buzz with excitement and caution, as AI accelerates discoveries once thought human-exclusive.
