mathematical

A mathematical problem had been resisting experts for more than 80 years. An AI has surpassed them all

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In 1946 the Hungarian mathematician Paul Erdős asked a seemingly very simple question: if you place n points in the plane, how many pairs of points can be exactly at a distance 1 from each other? This dilemma is known as unit distance problem in the planeand has maintained many mathematicians who research in the field of geometry, immersed in its resolution for no less than eighty years. The classic strategy proposed by many of them to try to solve it was to resort to a square grid. They soon realized that the number of pairs at unit distance grows at least as n to the power of (1 + C/loglog(n)), where C is a positive constant that quantifies how much a particular construction can be better than a basic square grid. It’s a complicated idea, it’s true, but we can try to approach it in a slightly more intuitive way. A standard square grid produces approximately 2n pairs of points at unit distance. If we rescale it in an ingenious way by choosing the scale factor as a number that has many divisors (in number theory this property is known as a number with many small prime factors), you get more pairs of points to fall exactly at distance 1. The value of C measures precisely the efficiency of that choice. This is the key. An AI from OpenAI has achieved the first major breakthrough in 80 years As we are seeing, the question Erdős asked is very easy to state, but extraordinarily difficult to resolve. If we develop the classical approach a little further we will realize that since loglog(n) grows very slowly, the exponent approaches 0. This means that the square grid grows only slightly faster than n, but not enough to exceed n at a fixed rate. This milestone was achieved by a general-purpose inference model that OpenAI was testing internally. This is why for decades mathematicians predicted that the upper bound would be approximately n^(1+o(1)), that is, just slightly larger than n. Now we know that they were wrong, and the person who refuted this conjecture was not a particularly skilled current mathematician; this milestone has pointed it out a general purpose inference model which OpenAI was testing internally. and not one artificial intelligence (AI) specialized in mathematics. This model has provided an infinite family of examples that produce polynomial improvement. In fact, he has shown that it is possible to construct configurations of points with at least n^(1+δ) pairs at unit distance, where δ is a fixed value greater than 0 that does not disappear as n grows. When the AI ​​delivered this result, OpenAI researchers asked a group of Princeton mathematicians to review it. And his conclusion was blunt. The AI ​​was right. This is the first progress on the lower bound of the problem posed by Erdős in 80 years. And, curiously, the OpenAI model has achieved this by using advanced engineering tools. algebraic number theory for an apparently elementary geometry problem. Several renowned mathematicians, such as Fields Medal winner Tim Gowers or number theory expert Arul Shankar, have declared that the result that AI has delivered is an extraordinary achievement that could provide mathematicians with a bridge to explore other problems in the future. Image | Jeswin Thomas More information | OpenAI In Xataka | These two problems have baffled mathematicians for decades. A genius has solved them with a stroke of the pen

A young man has solved a mathematical problem that lasted 60 years in 80 minutes with ChatGPT. That’s the least interesting thing about the story.

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He is 23 years old, his name is Liam Price and he has no advanced mathematical training. Even so, a few days ago he opened the Erdös problem websitepicked one at random and pasted it into ChatGPT. I didn’t know the history of the problem or who had tried it before. What he received back seemed like a good solution, and after consulting with a friend who was studying mathematics, the two realized they might be on to something special. A few hours later Terence Tao, one of the most renowned mathematicians in the world, confirmed that problem #1196 of Erdös, a conjecture about primitive sets of integers that had not been solved since 1966, had a solution. I had found her GPT-5.4 Pro in just 80 minutes. Not like that. This problem analyzed a question about the behavior of a particular mathematical sum on primitive sets, that is, sets of integers where none divides the other, when those numbers become very large. Jared Lichtman, a Stanford mathematician, had spent years on the problem and had made partial progress, but he and those who had tried before were starting from the same starting point that seemed like the right path. A novel idea. GPT-5.4 used another starting point. He stayed in the airmetic terrain and used a special function called von Mangoldt functiona classic tool of number theory known for its connections to prime numbers and Riemann zeta function. No one had thought about that approach to the problem, and as Lichtman explained when talking about the OpenAI model solution, “The LLM took a completely different route.” The achievement is real, but with nuances. Litchman praised the proposed solution by GPT-5.4, but there is one detail that has been omitted in many comments on this event: the raw output of ChatGPT was, in the words of this mathematician, “pretty poor.” This solution made it necessary for several experts to interpret it, detail it and extract from it the underlying idea that allowed the conjecture to be solved. Price didn’t know he had the solution until his friend read it, and he wasn’t sure until Tao confirmed it. The official repository of AI contributions to Erdös problemsmaintained by Tao himself on GitHub, classify the result as a solution generated in human-AI collaboration, not as a solution developed solely by AI. The distinction is important. A previous scandal. A few weeks ago Sebastien Bubeck, a researcher at OpenAI, posted on X that GPT-5 had “solved” several Erdös problems. That publication exceeded 100,000 views, but the mathematical community and also that surrounding the AI ​​industry criticized that statement. Demis Hassabis, CEO of DeepMind, called that statement “shameful.” What had actually happened is that the model I had found solutions to already solved problems on the web. Bubeck finished deleting the original tweet and tried to back down, but all this raised doubts about the validity of the application of AI to solve mathematical problems. AI and the mathematical success rate. Terence Tao and Nat Sothanaphan maintain the aforementioned record of all AI contributions to Erdös problems on GitHub. Each of the entries in that list or table is classified with a traffic light: green for complete solution, yellow for partial progress, and red for failure. In the category of completely AI-generated solutions with no known prior literature there are three green, fourteen yellow, and eight red traffic lights. However, the repository itself adds a unique comment: those who try to use AI to solve these problems and fail do not usually report it, so it is likely that AI has been applied “silently” to a large number of these problems without success, and those attempts do not appear in any table. There is a clear bias here because only successes generate headlines. Trying to measure what matters. In February 2026, eleven mathematicians created the initiative “First Proof“. In it they included ten mathematical problems that arose naturally in their research projects. For each one they included encrypted answers uploaded to a verification site, and they gave the AI ​​systems a week to try to solve those problems that had never appeared in any training data set. Preliminary results indicate that today AI models cannot overcome that barrier autonomously, and what happens is that there are still limits to what AI can really contribute in mathematics. But then, is there a revolution or not? Terence Tao offered a clear explanation as to why GPT-5.4 had succeeded where others had failed for 60 years. What had happened was what he described as a collective blockage of the mathematical community, because everyone started from the same origin because it was “the natural one”, the one marked by tradition. The AI ​​didn’t know that was the “correct” way to start, and that ignorance turned out to be an advantage. It’s not that the AI ​​was smarter, it’s that it had no biases about how to approach the problem. Now it remains to be seen if this novel way of trying to solve problems in unorthodox ways works. This will confirm whether what happened with Erdös’s problem number 1196 was an isolated case or whether a 23-year-old boy has managed to change our vision of how to tackle mathematical problems. Image | Universal Pictures In Xataka | There is a mathematically perfect way to cut a ham and cheese sandwich and it has been discussed since 1938.

The science of learning dismantles the mathematical rule of the fashionable study method

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When it comes to studying anything, almost all of us want to have a system that allows us learn quickly and efficient. This is where we can turn to the Internet, where there are numerous pages that promise us almost miraculous systems to pass easily, and one of them is the 2-7-30 method. But… What does science say about this system? What is it about? This method focuses on a system where you have to review the information exactly 2, 7 and 30 days after having addressed it on the first occasion. Something that is quite similar to what we want to achieve with the flashcards. Something that a priori seems quite simple to put into practice, but which can generate quite a bit of fear by leaving a topic shelved for so many days in the last round. It gives good results. But it is the best from the point of view of science, and to understand it, we have to go to the basics of how our memory works. And this method is based on the spacing effectwhich undoubtedly far surpasses the classic ‘binge’ the night before an exam, where you try to get all the data in in a matter of hours. Here, a classic meta-analysis published in 2006 in Psychological Bulletin, analyzed 839 measures in 317 experiments and confirmed that distributing practice over separate intervals dramatically improves retention. But even in the past, other studies suggested that repeating material over time consolidates memory much more efficiently. Recovery practice. There is no point in spacing out the reviews if, when day 2 or day 7 of the method arrives, we limit ourselves to passively rereading the notes. Here different studies have shown that actively trying to remember information produces much more lasting learning than passively re-studying it. In this way, forcing the brain to “rescue” that data strengthens neuronal connections, and science points to the advantage of active remembering over traditional binge-watching methods, such as making conceptual maps. The enemy to beat. The concept of reviewing in increasingly longer windows of time is born from the need to combat our natural decline in retention. This is where a work on the “forgetting curve” by Hermann Ebbinghaus comes into play, which demonstrated that we lose most of the newly learned information within hours or days if we do nothing to retain it. More modern replications of this idea confirm that this initial rapid forgetting is real and useful to contextualize the problem, although researchers depend on different factors and not only the strict passage of time. That is why the idea we should stick with is that every time we review the information, the forgetting curve resets and its slope becomes gentler so that it takes longer to disappear. The myth of exact numbers. Although it has been shown that spacing study days, in reality science does not identify 2, 7 and 30 days as a universally valid pattern for all learning and people, but will depend on many factors. Here, a study published in 2008 showed that the optimal interval between reviews depends on the retention interval we are looking for, but that the spacing changes radically if the objective is to remember something for an exam that is due in a week versus if we want to remember something in a year, as can happen in an opposition. In this way we get the following pattern: If the exam is in 1 week, the reviews should be separated by just 1 or 2 days. If the exam is in 1 year, Reviews should be spaced several weeks or even a month apart. Images | freepik In Xataka | SQ3R technique: the study method that helps you understand the subjects, not just remember them

A new mathematical proof settles the debate over whether the universe is a simulation

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What if everything we see, feel and experience is not real? It is one of the most fascinating ideas in science fiction and modern philosophy, in which it is proposed that everything around us it’s a real simulation of computer of some higher civilizationas if we were literally sims. And such is its magnitude, that science has had to come out to deny this idea. The problem. The ‘simulation hypothesis’ has gone beyond being a simple movie premise to a serious debate in technology circles and physical. The argument is usually statistical: if a civilization can create one simulation of reality, it will probably create many. These simulations could in turn generate their own and in this infinite ‘stack’ of realities, the odds that our universe be original, they are almost non-existent. And although this has been a very restrained topic among philosophers, science has also wanted to fully enter into research to respond to a problem within fundamental physics and pure mathematics. And the answer is quite clear: we are not in a simulation. The study. An international team of physicists, including Dr. Mir Faizal of the University of British Columbia (UBC) and renowned physicist Dr. Lawrence M. Krauss, has mathematically proven that the universe cannot be a computer simulation. His findings, published in it Journal of Holography Applications in Physicsnot only disprove the idea, but reveal something much deeper about the nature of reality: the universe is based on a type of “understanding” that exists beyond the reach of any algorithm. The reality. To understand this test, we must first understand what ‘reality’ is. Modern physics no longer sees the universe as tangible ‘matter’ moving in empty space, but thanks to Einstein space and time merged to now demonstrate that the microscopic world is probabilistic. The most widely accepted theory today focuses on quantum gravity, which suggests that space and time are fundamental. They are “emergent”: they spring from something deeper, something more like pure information. In this way, physicists assume that a “Theory of Everything“(ToE) that unifies gravity and quantum physics would, in essence, be a large axiomatic system: a set of meaningful rules and algorithmic calculations from which the entire universe, including spacetime itself, could be “computed” and generated. Incompleteness Theorems. In 1931, logician Kurt Gödel demonstrated something that blew up the foundations of mathematics: any formal system (such as a computer program or a set of physical laws) that is complex enough to include basic arithmetic will be incomplete or inconsistent. By ‘incomplete’ we mean that there will be true statements within the systems themselves that will never be able to be demonstrated following their own rules. It’s like the famous paradox that says “this statement is true, but it cannot be proven.” Faizal’s team argues that any purely algorithmic ToE would suffer from this limitation. There would always be “Gödelian truths” about the physics of the universe (perhaps about specific microstates of black holes or the nature of the singularity) that such a computational system could not test. Two layers. If the algorithmic universe is “incomplete”, how does our reality seem to work? Researchers propose that reality is not only the algorithm. This is what allows the universe to “know” that these Gödel truths are true, even though the algorithm alone cannot prove them. It is a fundamental layer of reality that transcends simple computing. The final test. With all the pieces on the table, the refutation of the simulation hypothesis becomes clear and elegant. The first of all is that every simulation is logarithmic, that is, a computer executes a problem following very specific rules that leave no room for doubt. In this way, we come face to face with our theories that are not ‘perfect’ in their demonstrations. But they don’t stop there, since scientists have pointed out that an algorithm can only simulate the algorithmic part, meaning that a computer could only, in the best of cases, emulate the computational and incomplete part of our universe. And the most important thing without a doubt is that our universe is more than an algorithm, since as Gödel’s theorems demonstrate, complete physical reality must include a non-algorithmic layer to be consistent and complete. Images | Compare Fiber In Xataka | Exactly 100 years ago we began to understand how the world works. Quantum physics has radically changed our lives

The end of mathematical problems without solution

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To the models of artificial intelligence (AI) Currently, mathematics are good. In fact, in October 2024 Goal AIthe finish line, managed to generalize Lyapunov’s function. The Russian mathematician Aleksander Lyapunov proposed the concept of the function that bears his name in 1892. His work is a very important tool in the study of dynamic systems, but mathematicians have since struggled to find a general method that allows identifying the functions of Lyapunov. And they have not been successful. However, Goal AI has had it. This is not at all the only recent success of AI models in the field of mathematics. Sergei Gukov, Professor of Theoretical Physics and Mathematics at the California Institute of Technology (Caltech), leads a team of researchers who are looking for The way to use this technology to solve advanced mathematical problems that require thousands, millions, or even billions of steps. Currently these scientists are working on the conjecture of Andrews-Curtis, a problem of combinatorial theory of groups proposed 60 years ago. Google and Openai AI have won gold in the mathematics Olympiad Gukov and his team have not yet managed to solve the main conjecture, but with the help of AI they have achieved something important: they have refuted several families of problems related to the conjecture of Andrews-Curtis and known as counterexamples that have remained open for more than 25 years. Gukov acknowledges That current AI models have important limitations when facing very complex mathematical problems, but it has the hope that in the future this technology allows the human being to solve Mathematical Millennium Problems. The best asset that researchers have to face this challenge is to instruct AI by resorting to learning for reinforcement According to this mathematician, the best asset that researchers have to face this challenge is to instruct AI by resorting to Reinforcement learning. Anyway, something important has just happened. As we anticipate in the holder of this article, the models of Google and Openai AI They have won the gold in The International Mathematics Olympiad. Both managed to solve five of the six problems posed using general purpose reasoning models capable of processing mathematical concepts using natural language. This strategy is different from the one they have previously used the AI companies in mathematical tests. In any case, According to SCMP An expert they have consulted argues that the speed with which the AI models are being developed suggests that they are less than a year after being used to solve some mathematical problems that still have no solution. As we have seen, Sergei Gukov defends this same idea, although this last mathematician has not dared to specify the moment in which AI will begin to solve the problems in which mathematicians have been engaged decades. Who knows, perhaps the solution to the millennium problems is close. Hopefully. Image | Jesus Thomas More information | SCMP In Xataka | These two problems have baffled mathematicians for decades. A genius has solved them with a stroke

AI is already our best ally to solve the mathematical problems that seem impossible

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The applications of the artificial intelligence (AI) are presumably unlimited. Beyond the daily uses with which many of us are already familiar with the design of drugs, Disease diagnosisthe OPTIMIZATION OF INDUSTRIAL PROCESSES or the Analysis of physical or chemical mechanisms complexes, among other options. It is even being used to solve mathematical problems of enormous difficulty. In addition, algorithms that use deep neuronal networks and Automatic learning They are designed to identify complex patterns in large volumes of information, which allows them to recognize images, voice or process natural language greatly. AI has reached our lives, and it is clear that it will stay, but the most surprising thing is that it is consolidating as an extremely valuable tool in relatively exotic fields. It is possible that AI helps us solve the mathematical problems of the millennium In October 2024 Goal AIMeta’s artificial intelligence, managed to generalize Lyapunov’s function. The Russian mathematician Aleksander Lyapunov proposed the concept of the function that bears his name in 1892. His work is a very important tool in the study of dynamic systems, but mathematicians have since struggled to find a general method that allows identifying the functions of Lyapunov. And they have not been successful. However, goal has had it. Our mathematical knowledge will no longer be limited by intuition and human capacity The strategy used by the company led by Mark Zuckerberg to solve the challenge of Lyapunov’s functions has consisted of training an AI model to recognize patterns and relationships between certain dynamic systems and its corresponding functions of Lyapunov. This is precisely what is good for AI. And it is a huge success because our mathematical knowledge will no longer be limited for intuition and human capacity. The AI ​​puts in our hands a new way of addressing complex mathematical problems, identifying patterns that a priori remain hidden for the human being. However, in the field of mathematics AI still has to improve to help us solve the great challenges that the human being has ahead. Sergei Gukov, professor of theoretical physics and mathematics at the California Institute of Technology (Caltech), leads a team of researchers who is looking for ways to use AI to solve advanced mathematical problems that require thousands, millions, or even billions of steps. These scientists are currently working on The conjecture of Andrews-Curtisa group combinatorial theory proposed 60 years ago. They have not yet managed to solve the main conjecture, but with the help of AI they have achieved something important: they have refuted several families of problems related to the conjecture of Andrews-Curtis and known as counterexamples that have remained open for more than 25 years. Gukov acknowledges that Current AI models have important limitations When facing very complex mathematical problems, but you hope that in the future this technology allows the human being to solve Mathematical Millennium Problems. According to this mathematician, the best asset that researchers have to face this challenge is to instruct AI by resorting to Reinforcement learning. Image | Generated by Xataka with Dall-e More information | IEEE Spectrum In Xataka | These two problems have baffled mathematicians for decades. A genius has solved them with a stroke

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