In 1966 Paul Erdös, one of the most prolific mathematicians in history, formulated a problem – one of the Erdos problems – concerning a particular set of numbers called the “primitive set”. For decades many scholars have tried to solve it, obtaining only partial results. In short, for 60 years it remained unsolved. In April 2026 Liam Price, a mathematics enthusiast who often tests the abilities of AI by submitting it unproven theorems, decided to also submit this problem to ChatGPT. With a single prompt and 80 minutes and 17 seconds processing, the system returned a convincing demonstration which now, thanks to the contribution of professional mathematicians, has been expanded, published and used to prove other theorems.
The solution proposed by AI is based on a mathematical tool already known and used in other contexts. AI, therefore, he did not create from nothing a new demonstration, but it connected existing techniques in an unexpected way. The fact that AI has managed to solve this problem is remarkable, but it does not mean that it is now capable of tackling every open mathematical question. Rather, it shows what these systems can become useful tools in researchespecially to indicate new uses of known techniques.
Let’s see what Erdös’ problems are, which problem was solved and how.
What are Erdös problems and what problem has been solved
Erdos’ problems are conjectures open, i.e. theorems formulated but not yet proven, proposed by mathematician Paul Erdös throughout his incredible career. Erdös was one of the most prolific and eccentric mathematicians in history. He has published around 1500 scientific articles, working and solving problems in very different areas of mathematics: graph theory, number theory, set theory, analysis and even probability. Precisely for this reason, Erdös’s problems touch on many different areas and degrees of difficulty, ranging from simple curiosities to central problems for some branches of mathematics.
The Erdös problem number 1196solved by ChatGPT and Liam Price, concerns special sets of numbers called “primitive setsIn these sets, no number is a divisor of the others. That is, taking any pair of numbers from a primitive set and dividing one element by another, we always obtain a number with a decimal point.
For example, 4, 6, 9 and 15 form a “primitive set”, because if we take two of these numbers (for example 15 and 4) and try to do a division, we get 15:4 = 3.75. The numbers 5, 6, 9 and 15, however, do not form a primitive set, because 5 is a divisor of 15.
Erdös’ conjecture concerned a particular value, called Erdös sumof the elements of a primitive set. According to the conjecture, when the numbers of the primitive set are sufficiently large this sum cannot be worth more than 1.
This conjecture, proposed in 1966, has been analyzed and studied by numerous famous mathematicians. Over the years, important progress has been achieved, but a complete demonstration was still missing.
How the conjecture was solved and the developments of the proof
In April 2026, 23-year-old Liam Price solved the conjecture thanks to a single prompt given to ChatGPT. In this prompts, asked:
Don’t search on the Internet. This is a test to evaluate your ability to develop non-trivial, original and creative proofs starting from a mathematics problem related to “number theory and primitive sets”. Provide a complete proof or disproof of the problem.
He then pasted the text of the conjecture, also providing the previous partial solutions, with the indication:
Information that may perhaps be useful to you in solving the problem described above.
ChatGPT, and in particular the model GPT 5.4he processed the request for 80 minutes and 17 seconds and then proposed a proof to the conjecture, which seemed correct in every way. After the first check done with their friend and mathematics student Kevin Barreto, the two proceeded to contact experts, including Terence Taoone of the most brilliant living mathematicians and winner of the Fields Medal. With expert support, the proposed solution was revised, expanded and used to improve some existing proofs and to solve two new conjectures. The results of this work can already be found on arXiv, the platform that collects the scientific articles awaiting review. Liam Price and Kevin Barreto were also included as authors in the article.
It is important to note, however, that the solution found by AI is not entirely new and original. To solve the conjecture, the AI used a mathematical object called a “descending von Mangoldt chain,” which has already been used in similar problems, but never applied in this specific way. AI, therefore, was capable of reorganize existing knowledgecombine known techniques and indicate new possible uses, but is not yet capable of creating true mathematical innovations in a completely autonomous manner.
