\n<\/p>\n
One idle evening last October, Mehtaab Sawhney took up an old pastime. He began perusing the website erdosproblems.com, an updated record of the 1,179 conjectures left behind by the eccentric and indefatigable 20th-century mathematician Paul Erdős.<\/p>\n
Sawhney, a mathematician at Columbia University, had always been interested in the Erdős problems, which range from minor curiosities to central open problems in number theory and combinatorics.<\/p>\n
He came upon a problem, #339, that seemed too straightforward to still be \u201copen\u201d nearly two decades after Erdős\u2019s death. He\u2019d seen similar conjectures before. \u201cThere were a number of problems that kind of looked too approachable,\u201d Sawhney says. In the past, he\u2019d turned to Google. \u201cAnd then eventually, with enough searching, I would find a reference to a solution.\u201d<\/p>\n
If you’re enjoying this article, consider supporting our award-winning journalism by subscribing. By purchasing a subscription you are helping to ensure the future of impactful stories about the discoveries and ideas shaping our world today.<\/p>\n
But recently he\u2019d been playing with ChatGPT as a new way to check the literature. \u201cI decided to plug it in, and then it just told me there was a reference,\u201d Sawhney says.<\/p>\n
It went so well that he reached out to a fellow mathematician, Mark Sellke, who had recently gone on leave from an academic position to work for OpenAI. Together they prompted ChatGPT to dig up lost solutions to nine other Erdős problems, plus partial solutions to 11 more.<\/p>\n
Since then, the website\u2019s activity has skyrocketed. According to a webpage started by the mathematician Terence Tao, AI tools have helped transfer about 100 Erdős problems into the \u201csolved\u201d column since October. The bulk of this assistance has been a kind of souped-up literature search, as it was with Sawhney\u2019s initial success. But in many cases, LLMs have pieced together extant theorems\u2014often in dialogue with their mathematician prompters\u2014to form new or improved solutions to these niche problems. In at least two cases, an LLM was even able to construct an original and valid proof to one that had never been solved, with little input from a human.<\/p>\n
The story of the Erdős problems is just part of a sea change that has taken place over the past few months. LLMs have become unrivaled in their ability to scour and synthesize the literature on any mathematical topic, however esoteric. They can also guide working mathematicians, helping them sketch a path to proving a larger result and proving small chunks of it to save time. This assistance is often misguided and riddled with holes that require expert eyes to suss out. But mathematicians can see its potential.<\/p>\n
\u201cThey are now useful research assistants,\u201d says Andrew Sutherland, a mathematician at the Massachusetts Institute of Technology. \u201cMathematicians whose only experience with LLMs is with earlier models don\u2019t yet fully appreciate this.\u201d<\/p>\n
AI is still nowhere near being able to solve major open problems in math, let alone replace mathematicians. Despite widespread anxieties voiced by graduate students during conference coffee breaks and in online message boards, no major mathematics journal has published a peer-reviewed proof citing the use of LLMs. But that, at least, could change this year.<\/p>\n
Erdős problems are a useful LLM \u201cbenchmark\u201d because there are so many of them. And they\u2019ve proved a distinctive showcase for the technology\u2019s burgeoning strength as a mathematical search engine.<\/p>\n
\u201cErdős problems sort of fit in a category of their own,\u201d Sutherland says. \u201cFor the most part, they\u2019re individual problems whose solution is not necessarily going to have any broader implications.\u201d As a result, solving a more obscure Erdős problem is a feat that often goes unnoticed. It\u2019s rarely worth submitting to a journal and rarely cited in subsequent work.<\/p>\n
None of that matters to an LLM. It can easily unearth preprint papers unknown even to experts\u2014proofs that sometimes don\u2019t reference Erdős at all. Google\u2019s Gemini found an offhand remark deep in a paper from 1981 that unknowingly solved Erdős problem #1089. But more surprising is LLMs\u2019 ability to make meaningful mathematical suggestions.<\/p>\n
\u201cI think it\u2019s a mistake to say it\u2019s \u2018just a search engine,\u2019\u201d Sutherland says. \u201cI\u2019ve had one or two interactions where it actually pointed me to a result that let me prove something I was stuck on.\u201d<\/p>\n
Similar experiences motivated the team behind First Proof,, a fresh attempt to test AI\u2019s math skills. Eleven top mathematicians picked discrete chunks of proofs they have completed but not yet published and posed them as a challenge to AI last Thursday. The problems cover a wide range of areas and vary in complexity. \u201cA system that could resolve all of them would be very useful for a professional mathematician,\u201d says Daniel Litt, a mathematician at the University of Toronto.<\/p>\n
The team is giving LLMs until Friday to produce proofs of the 10 problems. The one-week time limit was chosen carefully, according to Lauren Williams, a Harvard University mathematician on the First Proof team. It\u2019s less time than her own problem took her and a coauthor to prove, so likely not long enough for human mathematicians without AI assistance.<\/p>\n
By Monday the e-mails and social media pages of Williams and her collaborators were inundated with claimed solutions. \u201cThere\u2019s a lot of excitement, which is really great to see,\u201d she says. A Discord server hosting discussions on the challenge has quickly garnered hundreds of members, many carrying purported proofs from ChatGPT and other LLMs.<\/p>\n
Familiar troubles have already arisen. First Proof was meant to be more than a literature search\u2014the team tested its questions on LLMs to be sure no answers existed in their training data. But pretty quickly an online solution surfaced to a problem from Martin Hairer, winner of a 2014 Fields Medal, math\u2019s highest honor\u2014and one of the First Proof team members. When he picked the problem, he had overlooked a partial proof in the bowels of his personal website that was archived by the Wayback Machine.<\/p>\n
And contestants lacking the team\u2019s expertise in these particular mathematical niches aren\u2019t sure what to do with the deluge of confident claims their LLMs keep spitting out\u2014it\u2019s up to the First Proof team to check every submission. \u201cVerification is a problem because 90 percent of the time it will come up with a solution,\u201d Williams says. \u201cIt\u2019s going to write something and sound confident about it.\u201d<\/p>\n
Litt has glanced over many of the \u201cproofs\u201d circulating this week and found them to be largely bogus\u2014although he\u2019s seen a few that may be correct. \u201cIt\u2019s absolutely very impressive that the models are sometimes able to generate correct answers to some of the problems,\u201d he says. \u201cBut they\u2019re generating a huge amount of garbage.\u201d Even by Saturday, it may not be clear whether the LLMs have won or lost.<\/p>\n
Regardless of the First Proof outcome, the last month has brought many signs that LLMs will soon be part of many mathematicians\u2019 tool chests.<\/p>\n
In January Ravi Vakil, current president of the American Mathematical Society, posted a preprint with two other mathematicians and two researchers from Google in which they collaborated to solve a math problem that bears on his research. The authors document how Google\u2019s LLM helped them get to a proof. \u201cIt really did lead us to new ideas,\u201d says Vakil, who wanted to \u201cget a sense of how mathematicians should reasonably be doing math in five years.\u201d<\/p>\n
Still, LLMs have yet to contribute a proof that would create buzz if it came from a human. \u201cEvery individual result has been vastly overhyped by certain corners of the Internet,\u201d Litt says. Carlo Pagano, who collaborated with Google\u2019s DeepMind team to work on several Erdős problems using Gemini in research posted as a preprint, is also hoping for a more substantial benchmark. \u201cThe Erdős problems are not great in some sense,\u201d he says. \u201cIt\u2019s important to do this also on problems that we know are of broader interest.\u201d<\/p>\n
But several mathematicians predicted that 2026 will be the year where results of this type, in which AI is a stated contributor, first make it through peer review in major mathematics journals.<\/p>\n
\u201cI think it\u2019s going to change the subject,\u201d Sawhney says. \u201cAnd that\u2019s a really exciting thing.\u201d Given that change, Sawhney has taken an academic leave from Columbia to work for OpenAI. This week Pagano started a joint position at Google DeepMind. \u201cIt\u2019s clear that this will change how we do math,\u201d he says, \u201cso better to start early rather than later.\u201d<\/p>\n","protected":false},"excerpt":{"rendered":"
One idle evening last October, Mehtaab Sawhney took up an old pastime. He began perusing the website erdosproblems.com, an updated record of the 1,179 conjectures left behind by the eccentric and indefatigable 20th-century mathematician Paul Erdős. Sawhney, a mathematician at Columbia University, had always been interested in the Erdős problems, which range from minor curiosities<\/p>\n","protected":false},"author":1,"featured_media":44366,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[58],"tags":[1208,23045,4693,2388,2196,7040,13120,15204],"class_list":{"0":"post-44365","1":"post","2":"type-post","3":"status-publish","4":"format-standard","5":"has-post-thumbnail","7":"category-science","8":"tag-closer","9":"tag-erdos","10":"tag-math","11":"tag-moving","12":"tag-problems","13":"tag-solutions","14":"tag-transforming","15":"tag-uncovers"},"_links":{"self":[{"href":"https:\/\/naijaglobalnews.org\/index.php?rest_route=\/wp\/v2\/posts\/44365","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/naijaglobalnews.org\/index.php?rest_route=\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/naijaglobalnews.org\/index.php?rest_route=\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/naijaglobalnews.org\/index.php?rest_route=\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/naijaglobalnews.org\/index.php?rest_route=%2Fwp%2Fv2%2Fcomments&post=44365"}],"version-history":[{"count":0,"href":"https:\/\/naijaglobalnews.org\/index.php?rest_route=\/wp\/v2\/posts\/44365\/revisions"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/naijaglobalnews.org\/index.php?rest_route=\/wp\/v2\/media\/44366"}],"wp:attachment":[{"href":"https:\/\/naijaglobalnews.org\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=44365"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/naijaglobalnews.org\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=44365"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/naijaglobalnews.org\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=44365"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}