O<\/span>n Could 20, 2026, OpenAI made an announcement that shook the mathematical world. An inner AI mannequin \u2014 one not out there to the general public \u2014 had give you a counterexample to the \u201cunit distance\u201d drawback, a conjecture made in 1946 by Paul Erd\u0151s, the prolific, itinerant Hungarian mathematician.<\/p>\n Erd\u0151s posed hundreds of questions, however this one was particular: It was each easy to clarify and mathematically deep. It was the primary traditionally important proof to return from an AI mannequin. Although the mannequin\u2019s end result wasn\u2019t definitive \u2014\u00a0human mathematicians would considerably enhance on it inside weeks \u2014 it was modern, bringing in concepts from a distant department of math that nobody had efficiently utilized to this drawback earlier than. And it was influential: Inside just a few days, associated strategies had been used to resolve different necessary issues.<\/p>\n Then on August 1, OpenAI introduced that an unreleased mannequin named Astra made 10 further mathematical advances, together with discovering options to 3 extra issues posed by Erd\u0151s.<\/p>\n Many mathematicians have hailed developments corresponding to these as a part transition within the mathematical functionality of AI fashions. These fashions are \u201caltering dramatically the best way mathematical analysis is being carried out,\u201d stated Noga Alon of Princeton College, who has solved dozens of Erd\u0151s issues over his decades-long profession.<\/p>\n Paul Erd\u0151s, one of the prolific mathematicians in historical past, was deeply whimsical when it got here to arithmetic, and deeply cynical when it got here to authority.<\/p>\n<\/p><\/div>\n Picture by George Csicsery from the documentary N is a Quantity: A Portrait of Paul Erd\u0151s \u00a91993. All Rights Reserved.<\/p>\n<\/div>\n<\/figcaption><\/figure>\n Erd\u0151s and his conjectures have lengthy fascinated mathematicians. He traveled consistently \u2014 residing out of a suitcase for years at a time, staying with pals, proudly owning nearly nothing. He rattled off issues in printed papers and letters to mathematicians all over the world, typically attaching prize cash that he would pay out of pocket to the primary particular person to give you an answer. The reward may be a token $10 or $25, or, for issues he thought-about necessary or tough, it may vary into the hundreds. Erd\u0151s died of a coronary heart assault in 1996 whereas attending a math convention in Warsaw, however a nonprofit basis primarily based in Iowa has promised to make good on his bounties.<\/p>\n He was a beloved determine, but additionally a downright bizarre one. He solely wore silk, and he averted the contact of different folks. Deeply cynical about authority, he gave away many of the cash he earned and relied on a pal to handle his funds and different sensible affairs. He referred to God because the \u201cSupreme Fascist\u201d and fueled his incessant output of mathematical concepts with a gentle food plan of amphetamines. It’s a unusual irony of historical past that the issues he steered have now turn out to be a central proving floor \u2014\u00a0and, in impact, a sequence of PR coups \u2014 for the world\u2019s largest and strongest expertise corporations.<\/p>\n However in all probability none of this might have occurred had it not been for an English mathematician named Thomas Bloom.<\/p>\n Like Erd\u0151s, Bloom was considering each quantity idea and combinatorics. His focus has been an space known as arithmetic combinatorics, which lies on the intersection of the 2. After getting his doctorate in 2014, Bloom established himself as a rising star within the area, touchdown a prestigious fellowship from Britain\u2019s Royal Society, which let him work at nearly any college he needed to. (He\u2019s now on the College of Manchester.)<\/p>\n Bloom has favored Erd\u0151s\u2019 type for so long as he can bear in mind. However he at all times discovered it arduous to maintain observe of which issues had been solved and which had been forgotten solely. So in early 2023, he determined to collect as many issues as he may into an inventory.<\/p>\n He meant it for his personal use. However \u201cI believed it will be simpler if I may entry it wherever I used to be,\u201d he stated; he figured he \u201cwould possibly as effectively make an internet site, form of with the expectation that possibly no person would use it.\u201d He gathered a pair hundred issues and launched erdosproblems.com. Bloom used ChatGPT to jot down the Python code that ran the web site, which was, on the time, a exceptional factor for a big language mannequin to have the ability to do. Utilizing one to collaborate on the maths itself nonetheless appeared like solely a distant chance.<\/p>\n Thomas Bloom\u2019s web site of Erd\u0151s issues grew to become a house for arithmetic at its greatest. Then AI got here on the scene.<\/p>\n<\/p><\/div>\n<\/figcaption><\/figure>\n His aim was not simply to cross gadgets off an inventory. He puzzled if \u201cmodern-day arithmetic, typically utilizing strategies unknown by Erd\u0151s, may clear up many of those extra obscure issues,\u201d he wrote in a weblog put up. \u201cWe are going to then be left with a core of fascinating, tough issues, which may serve to show the boundaries of our information.\u201d<\/p>\n Bloom did essential work in curating the checklist:\u00a0Typically Erd\u0151s said issues in ambiguous or unclear methods, and Bloom found out what essentially the most smart model of every drawback must be. He stored including issues to the positioning, and step by step its viewers grew. Over the course of 2024 and the primary eight months of 2025, the statuses of 111 issues on the checklist had been modified from \u201copen\u201d to \u201csolved\u201d (though a few of these had been solved years earlier, and their standing change mirrored the rediscovery or verification of a proof).<\/p>\n Then, in August 2025, some colleagues steered that Bloom add a commenting perform, so that individuals may speak about issues they had been considering. He was ready to take action shortly, utilizing ChatGPT to jot down the code. By now he\u2019d cataloged almost 1,000 issues.<\/p>\n Bloom\u2019s timing was good. He made it attainable for like-minded folks to speak to at least one one other, and that \u201cactually let a group construct up,\u201d he stated. For essentially the most half, feedback had been sporadic \u2014 an issue would possibly entice a single remark stating an instance or noting how arduous the issue regarded. However exercise steadily grew, and a few issues catalyzed nuanced mathematical discussions between strangers.<\/p>\n \u201cTom most likely by no means actually realized this, however for me it\u2019s actually modified my life,\u201d stated Wouter van Doorn, the fourth-most-prolific commenter on Bloom\u2019s web site. Like many individuals who grew to become energetic on the positioning within the autumn of 2025, van Doorn isn\u2019t precisely knowledgeable mathematician. He works \u201cfor an organization that will get employed by different corporations to do customer support assist,\u201d as he put it. However he isn\u2019t precisely an novice both \u2014 a decade prior, he nearly accomplished a grasp\u2019s diploma in math at KU Leuven in Belgium. In 2024, spurred partly by how succesful he noticed LLMs getting, he took a six-month depart of absence from work to concentrate on math. On the time, whereas he didn\u2019t notably wish to use AI, he remembers pondering, \u201cProper now I\u2019m nonetheless higher at arithmetic than an AI is, however who is aware of what it\u2019ll be in a 12 months, two years, 5 years? If I wish to end these tasks, and I need them to be mine, now could be the time.\u201d<\/p>\n \n Downside 1102<\/span>\n <\/p>\n We are saying that $latex A subseteq mathbb{N}$ has property $latex P$ if, for all $latex n geq 1$, there are solely finitely many $latex a in A$ such that $latex n + a$ is square-free. We are saying that $latex A$ has property $latex Q$ if there are infinitely many $latex n$ such that $latex n + a$ is square-free for all $latex a < n$. How briskly should sequences $latex A = {a_1 < a_2 < cdots}$ with properties $latex P$ or $latex Q$ enhance?<\/p>\n<\/div>\n And so, in October 2025, van Doorn, now again at his day job, left the primary touch upon the web page for Downside 1102. The issue, which Erd\u0151s posed in 1981, asks about properties of units of \u201csquare-free\u201d integers \u2014 that’s, integers that don’t have any repeated prime components. (As an example, 30 is square-free as a result of it is the same as 2 \u00d7 3 \u00d7 5, however 18 isn’t, as a result of it is the same as 2 \u00d7 3 \u00d7 3; the three repeats.)<\/p>\n In early November, van Doorn shared progress towards a solution \u2014 which he\u2019d found out with out counting on AI \u2014 as a touch upon the issue web page.<\/p>\n Later that day, one other commenter on the positioning replied, claiming he had discovered a flaw in van Doorn\u2019s argument. The 2 traded remarks in speedy succession, and van Doorn satisfied his interlocutor that his argument was right. \u201cI see how your argument works now. Good!\u201d the opposite mathematician replied. That different mathematician was Terence Tao, a professor on the College of California, Los Angeles who’s arguably the best-known mathematician alive at this time, and inarguably one of the influential. (Not by the way, when Tao was simply 10 years previous, he crossed paths with Erd\u0151s.)<\/p>\n Bloom\u2019s web site, which has the feel and appear of an earlier time, was turning into an instance of the web at its democratic greatest. \u201cThis complete collaboration wouldn’t have been attainable with out Tom\u2019s web site and the feedback part there,\u201d van Doorn stated. It didn\u2019t matter when you had tenure or not, when you had been younger or previous, when you had been at a elaborate college and even at a college in any respect. In case you needed to work on math and had good concepts, you possibly can discover folks to collaborate with.<\/p>\n However because the winter set in \u2014 across the similar time that van Doorn discovered himself collaborating with Terry Tao \u2014 issues began to vary.<\/p>\n Kevin Barreto and Liam Value, each of their early 20s, grew to become pals in the summertime of 2025 on a Discord server devoted to AI. Barreto is at the moment an undergraduate on the College of Cambridge; Value studied some math in faculty however left earlier than ending. In December, satisfied that the most recent AI fashions would possibly achieve resolving some Erd\u0151s issues, the pair began throwing batches of issues at them. They realized early on that in the event that they instructed GPT-5.2 that an issue\u2019s reply wasn\u2019t identified, it wouldn\u2019t make a lot headway, in order Barreto put it, they discovered how one can \u201cimmediate it in a really specific method, gaslighting it into pondering the issue is simpler than it truly is.\u201d<\/p>\n \n Downside 333<\/span>\n <\/p>\n Let $latex A subseteq mathbb{N}$ be a set of density zero. Does there exist a $latex B$ such that $latex A subseteq B + B$ and $latex |B cap {1, ldots, N}| = o(N^{1\/2})$ for all giant $latex N$?<\/p>\n<\/div>\n They’d what they thought was their first triumph on Erd\u0151s Downside 333. Early on Christmas morning, Barreto posted a proof to Bloom\u2019s web site, writing, \u201cWe consider, to the most effective of our information, that is the primary case of an LLM totally autonomously resolving an Erd\u0151s drawback, not beforehand resolved by people.\u201d Though 333, which handled the sums of units of integers, was not a very necessary drawback, fixing it with AI nonetheless felt necessary.<\/p>\n However just a few hours later, one other consumer identified that Erd\u0151s himself had offered a decision to 333 in a paper printed in 1977. Barreto owned as much as the error. \u201cMy formal request to all members of the web site is to place higher concentrate on literature search on the issues at the moment marked as open,\u201d he wrote. \u201cAs somebody who has fallen for this twice now, it\u2019s fairly gut-wrenching.\u201d<\/p>\n \n Downside 728<\/span>\n <\/p>\n Let $latex C > 0$ and $latex epsilon > 0$ be small enough. Are there infinitely many integers $latex a, b, n$ with $latex a geq epsilon n$ and $latex b geq epsilon n$ such that $latex a!b! mid n!(a + b \u2013 n)!$ and $latex a + b > n + C log n$?<\/p>\n<\/div>\n Undeterred, he and Value stored at it, and by January 4, 2026, they\u2019d used GPT-5.2 Professional to discover a resolution to Erd\u0151s 728, an issue about when sure numbers are divisible by different numbers. This time no person may discover prior work already proving it. Barreto used one other AI software known as Aristotle (developed by a startup known as Harmonic) to certify that the proof held collectively logically. Nat Sothanaphan, a software program engineer and the one discussion board participant extra prolific than Bloom, Tao, and van Doorn, had ChatGPT write up the formalized end result and posted it on-line.<\/p>\n Value developed a technique for how one can ask LLMs to resolve open questions. First, he would ask a chatbot for an answer. Then he would feed that resolution right into a recent occasion of the chatbot, asking it to examine the earlier chatbot\u2019s work. He\u2019d repeat this course of till he had what regarded like a workable resolution. (This echoes among the work that corporations have been doing internally to create what they name harnesses or scaffolds, which automate the type of iteration that Value does by hand.)<\/p>\n Barreto and Value\u2019s papers symbolize only a fraction of the numerous Erd\u0151s issues solved no less than partly by AI over the previous few months. There are a number of the reason why these issues specifically have turn out to be such a fertile take a look at mattress for LLMs. The first one is that, by and huge, Erd\u0151s issues are in quantity idea, combinatorics, and graph idea, all areas of math which have proved extra accessible than others to giant language fashions. The issues additionally fluctuate extensively in issue and mathematical significance. This variation makes them applicable for a nascent expertise whose talents additionally fluctuate extensively.<\/p>\n Lots of Erd\u0151s\u2019 issues had a financial worth connected to them from their second of inception, a playful incentive from a wandering eccentric. However now, as the issues have turn out to be a casual benchmark for AI, their options are being mentioned when it comes to their \u201cper-problem value\u201d \u2014 the worth of the tokens wanted to resolve them.<\/p>\n<\/p><\/div>\n Picture by George Csicsery from the documentary N is a Quantity: A Portrait of Paul Erd\u0151s \u00a91993. All Rights Reserved.<\/p>\n<\/div>\n<\/figcaption><\/figure>\n \u201cNumerous my current papers must be principally credited to AI,\u201d van Doorn stated. \u201cThe concepts concerned had been concepts I didn’t give you myself.\u201d Like many individuals energetic on the Erd\u0151s web site, van Doorn is worked up about the best way LLMs are permitting him to do extra issues extra shortly. \u201cIf I learn an concept by an LLM, I digest it, attempt to perceive it, simplify it, and generalize it,\u201d he stated. He makes use of AI to higher perceive the maths.<\/p>\n Not everybody holds themselves to this commonplace. \u201cAn enormous drawback is AI is getting used quite a bit by individuals who aren\u2019t mathematicians, who don\u2019t have an enormous mathematical background and are usually not able to verifying the output,\u201d Bloom stated. \u201cThey like to maneuver quick, ask their AI to examine it, it grows and grows. We\u2019re seeing much more of those 100- to 200-page papers that persons are posting. \u2018I solved this theorem; I obtained AI to generate the proof and examine the proof and write the paper.\u2019 However no human has learn it, and no human goes to learn it. It\u2019s an enormous problem now.\u201d<\/p>\n \n Downside 1196<\/span>\n <\/p>\n Is it true that, for any $latex x$, if $latex A subset [x, infty)$ is a primitive set of integers (so that no distinct elements of $latex A$ divide each other) then $latex displaystylesum_{a in A} frac{1}{a log a} < 1 + o(1),$ where the $latex o(1)$ term $latex to 0$ as $latex x to infty$?<\/p>\n<\/div>\n By Price\u2019s own assessment, he doesn\u2019t have enough mathematical understanding to verify the solutions he ultimately coaxes from the LLMs. But with Barreto\u2019s help, he\u2019s been able to find mathematicians knowledgeable and willing enough to\u00a0check the results. Both Price and Barreto are co-authors with Tao, Jared Duker Lichtman of Stanford University, and other accomplished mathematicians on a May 2026 paper resolving Erd\u0151s Problem 1196, one of their more significant results. (1196 asks about the possible size of so-called primitive sets \u2014 collections of integers, such as {2, 5, 9, 21}, in which no number divides any other.)<\/p>\n Bloom was surprised that despite lots of attention from OpenAI, Google DeepMind, and several startups, most of the new results had come from hobbyists and undergraduates using publicly available LLMs, not from corporate labs using more advanced internal models.<\/p>\n But that would change a few weeks later, on May 20, 2026, when OpenAI announced that they had solved one of the most well known Erd\u0151s problems of all, the unit distance problem.<\/p>\n In the first months of 2026, the major tech companies began to see opportunity in erdosproblems.com. As Lichtman explained, \u201cErd\u0151s had over 1,000 papers. They were scattered.\u201d An institute in Hungary had collected scanned images of many of the papers, but nobody had collected all the problems. \u201cThis kind of single repository that anyone can access \u2014 labs realized that this could effectively be a benchmark.\u201d<\/p>\n In January, a team of 24 researchers led by Google DeepMind shared a paper solving four problems and finding old, forgotten solutions to nine more, after \u201cusing Gemini to systematically evaluate 700 conjectures labeled \u2018Open\u2019 in Bloom\u2019s Erd\u0151s Problems database.\u201d In May, a separate DeepMind team of 21 researchers announced that \u201cour most capable agent autonomously resolved 9 of 353 open Erd\u0151s problems at the per-problem cost of a few hundred dollars.\u201d (As of this article\u2019s publication, Bloom\u2019s database contains 565 solved problems and 652 open ones, but the DeepMind team narrowed their search to problems that have been written in formal logic.)<\/p>\n And, on May 20, OpenAI shared a solution to the unit distance problem, along with a blog post explaining the work and a companion paper that featured nine world-class mathematicians commenting on the correctness of the proof and the importance of what had been done (as well as presenting a streamlined human version of the result). Mathematicians had generally believed that Erd\u0151s\u2019 conjecture \u2014 about how many evenly spaced points can be placed on a plane \u2014 was correct. To general surprise, OpenAI\u2019s internal model found a counterexample. To do so, it had found a sophisticated way to use tools from an area of math called algebraic number theory. As Jacob Tsimerman of the University of Toronto wrote in the companion article, \u201cThis is a really impressive piece of work. \u2026 It is definitely an intimidating construction.\u201d<\/p>\n In the same article, Tim Gowers of Cambridge and the Coll\u00e8ge de France wrote that \u201cif a human had written the paper and submitted it to the Annals of Mathematics and I had been asked for a quick opinion, I would have recommended acceptance without any hesitation. No previous AI-generated proof has come close to that.\u201d<\/p>\n The author on the paper that presented the original solution was given simply as \u201cOpenAI.\u201d<\/p>\n Terence Tao was 10 years old when he met Erd\u0151s.<\/p>\n<\/p><\/div>\n<\/figcaption><\/figure>\n Later, using techniques related to the ones the AI model had applied to the unit distance problem, a group of four mathematicians, including Bloom, disproved a version of another long-standing Erd\u0151s conjecture. The \u201csum-product\u201d conjecture proposed that if you have sets of numbers, either their sum or their product must grow quickly. The mathematicians found a set of real numbers for which both the sum and the product grow more slowly than expected. The conjecture for integers remains open.<\/p>\n Figuring out what impact AI will have on math and mathematicians means not only looking to its most important results, but also examining how it changes the everyday practice of solving quotidian problems. Noga Alon, the Princeton mathematician, estimates that he has solved a few dozen Erd\u0151s problems over his career. He has now stopped trying. \u201cOnce AI started to solve them, there is no point anymore,\u201d he said. Terry Tao has stepped away from the Erd\u0151s problem community to focus on getting work done.<\/p>\n Van Doorn, who for now still has his day job at a customer service company, said that LLMs \u201care clearly better at thinking and doing math than I am. I don\u2019t hold a candle to current AI systems.\u201d However, he added, \u201cthe eventual proofs that I write are simpler, more general, and easier to read for other people than the thing that ChatGPT came up with.\u201d AI has indisputably boosted his productivity, and he\u2019s still having fun. \u201cIf you want to play piano, you aren\u2019t going to hire a piano-playing machine that does it better than you. You will play the piano because you like playing the piano. I enjoy thinking about numbers, doing math, writing papers. I\u2019m not going to hire a paper-making machine that does it for me.\u201d<\/p>\n For van Doorn, there is joy to be found in digesting the responses he gets from LLMs. \u201cI\u2019ve been doing a lot of math recently thanks to the Erd\u0151s-problems community. It used to be the case I just did everything all by myself, struggled alone in my room. I don\u2019t know how it happened, but nowadays people contact me saying, \u2018I have this idea. Do you want to join me thinking about this?\u2019\u201d<\/p>\n The increasing capability of AI has made it easier for people like Price or van Doorn, with less mathematical training, to solve puzzles, while making those puzzles less interesting to people like Alon who have devoted a lifetime to understanding them.<\/p>\n![\"Black-and-white](\"https:\/\/www.quantamagazine.org\/wp-content\/uploads\/2026\/08\/Erdos-1-cr.Simons-Foundation.webp\")
<\/div>Many Conferences<\/h2>\n
![\"\"](\"https:\/\/www.quantamagazine.org\/wp-content\/uploads\/2026\/08\/Thomas-Bloom-cr.Thomas-Bloom.webp\")
<\/div>Journey to the Cross-Roads<\/h2>\n
![\"Black-and-white](\"https:\/\/www.quantamagazine.org\/wp-content\/uploads\/2026\/08\/Erdos-2-cr.Simons-Foundation.webp\")
<\/div>Many Partings<\/h2>\n
![\"\"](\"https:\/\/www.quantamagazine.org\/wp-content\/uploads\/2026\/08\/Unit-distance-box-V2.svg\")
\n <\/div>\n<\/figure>\n![\"\"](\"https:\/\/www.quantamagazine.org\/wp-content\/uploads\/2026\/08\/Paul-Erdos-with-Terence-Tao-cr-Billy_Grace-Tao-1.webp\")
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