Latest news with #mathematics
Yahoo
15 hours ago
- Science
- Yahoo
Mathematicians discover a completely new way to find prime numbers
When you buy through links on our articles, Future and its syndication partners may earn a commission. For centuries, prime numbers have captured the imaginations of mathematicians, who continue to search for new patterns that help identify them and the way they're distributed among other numbers. Primes are whole numbers that are greater than 1 and are divisible by only 1 and themselves. The three smallest prime numbers are 2, 3 and 5. It's easy to find out if small numbers are prime — one simply needs to check what numbers can factor them. When mathematicians consider large numbers, however, the task of discerning which ones are prime quickly mushrooms in difficulty. Although it might be practical to check if, say, the numbers 10 or 1,000 have more than two factors, that strategy is unfavorable or even untenable for checking if gigantic numbers are prime or composite. For instance, the largest known prime number, which is 2136279841 − 1, is 41,024,320 digits long. At first, that number may seem mind-bogglingly large. Given that there are infinitely many positive integers of all different sizes, however, this number is minuscule compared with even larger primes. Furthermore, mathematicians want to do more than just tediously attempt to factor numbers one by one to determine if any given integer is prime. "We're interested in the prime numbers because there are infinitely many of them, but it's very difficult to identify any patterns in them," says Ken Ono, a mathematician at the University of Virginia. Still, one main goal is to determine how prime numbers are distributed within larger sets of numbers. Recently, Ono and two of his colleagues — William Craig, a mathematician at the U.S. Naval Academy, and Jan-Willem van Ittersum, a mathematician at the University of Cologne in Germany — identified a whole new approach for finding prime numbers. "We have described infinitely many new kinds of criteria for exactly determining the set of prime numbers, all of which are very different from 'If you can't factor it, it must be prime,'" Ono says. He and his colleagues' paper, published in the Proceedings of the National Academy of Sciences USA, was runner-up for a physical science prize that recognizes scientific excellence and originality. In some sense, the finding offers an infinite number of new definitions for what it means for numbers to be prime, Ono notes. At the heart of the team's strategy is a notion called integer partitions. "The theory of partitions is very old," Ono says. It dates back to the 18th-century Swiss mathematician Leonhard Euler, and it has continued to be expanded and refined by mathematicians over time. "Partitions, at first glance, seem to be the stuff of child's play," Ono says. "How many ways can you add up numbers to get other numbers?" For instance, the number 5 has seven partitions: 4 + 1, 3 + 2, 3 + 1 + 1, 2 + 2 + 1, 2 + 1 + 1 + 1 and 1 + 1 + 1 + 1 + 1. Yet the concept turns out to be powerful as a hidden key that unlocks new ways of detecting primes. "It is remarkable that such a classical combinatorial object — the partition function — can be used to detect primes in this novel way," says Kathrin Bringmann, a mathematician at the University of Cologne. (Bringmann has worked with Ono and Craig before, and she's currently van Ittersum's postdoctoral adviser, but she wasn't involved with this research.) Ono notes that the idea for this approach originated in a question posed by one of his former students, Robert Schneider, who's now a mathematician at Michigan Technological University. Ono, Craig and van Ittersum proved that prime numbers are the solutions of an infinite number of a particular type of polynomial equation in partition functions. Named Diophantine equations after third-century mathematician Diophantus of Alexandria (and studied long before him), these expressions can have integer solutions or rational ones (meaning they can be written as a fraction). In other words, the finding shows that "integer partitions detect the primes in infinitely many natural ways," the researchers wrote in their PNAS paper. George Andrews, a mathematician at Pennsylvania State University, who edited the PNAS paper but wasn't involved with the research, describes the finding as "something that's brand new" and "not something that was anticipated," making it difficult to predict "where it will lead." Related: What is the largest known prime number? The discovery goes beyond probing the distribution of prime numbers. "We're actually nailing all the prime numbers on the nose," Ono says. In this method, you can plug an integer that is 2 or larger into particular equations, and if they are true, then the integer is prime. One such equation is (3n3 − 13n2 + 18n − 8)M1(n) + (12n2 − 120n + 212)M2(n) − 960M3(n) = 0, where M1(n), M2(n) and M3(n) are well-studied partition functions. "More generally," for a particular type of partition function, "we prove that there are infinitely many such prime detecting equations with constant coefficients," the researchers wrote in their PNAS paper. Put more simply, "it's almost like our work gives you infinitely many new definitions for prime," Ono says. "That's kind of mind-blowing." The team's findings could lead to many new discoveries, Bringmann notes. "Beyond its intrinsic mathematical interest, this work may inspire further investigations into the surprising algebraic or analytic properties hidden in combinatorial functions," she says. In combinatorics — the mathematics of counting — combinatorial functions are used to describe the number of ways that items in sets can be chosen or arranged. "More broadly, it shows the richness of connections in mathematics," she adds. "These kinds of results often stimulate fresh thinking across subfields." Bringmann suggests some potential ways that mathematicians could build on the research. For instance, they could explore what other types of mathematical structures could be found using partition functions or look for ways that the main result could be expanded to study different types of numbers. "Are there generalizations of the main result to other sequences, such as composite numbers or values of arithmetic functions?" she asks. "Ken Ono is, in my opinion, one of the most exciting mathematicians around today," Andrews says. "This isn't the first time that he has seen into a classic problem and brought really new things to light." RELATED STORIES —Largest known prime number, spanning 41 million digits, discovered by amateur mathematician using free software —'Dramatic revision of a basic chapter in algebra': Mathematicians devise new way to solve devilishly difficult equations —Mathematicians just solved a 125-year-old problem, uniting 3 theories in physics There remains a glut of open questions about prime numbers, many of which are long-standing. Two examples are the twin prime conjecture and Goldbach's conjecture. The twin prime conjecture states that there are infinitely many twin primes — prime numbers that are separated by a value of two. The numbers 5 and 7 are twin primes, as are 11 and 13. Goldbach's conjecture states that "every even number bigger than 2 is a sum of two primes in at least one way," Ono says. But no one has proven this conjecture to be true. "Problems like that have befuddled mathematicians and number theorists for generations, almost throughout the entire history of number theory," Ono says. Although his team's recent finding doesn't solve those problems, he says, it's a profound example of how mathematicians are pushing boundaries to better understand the mysterious nature of prime numbers. This article was first published at Scientific American. © All rights reserved. Follow on TikTok and Instagram, X and Facebook.
New York Times
2 days ago
- Science
- New York Times
Can A.I. Quicken the Pace of Math Discovery?
Artificial intelligence can write a poem in the style of Walt Whitman, provide dating advice and suggest the best way to cook an artichoke. But when it comes to mathematics, large language models like OpenAI's immensely popular ChatGPT have sometimes stumbled over basic problems. Some see this as an inherent limitation of the technology, especially when it comes to complex reasoning. A new initiative from the Defense Advanced Research Projects Agency, or DARPA, seeks to account for that shortfall by enlisting researchers in finding ways to conduct high-level mathematics research with an A.I. 'co-author.' The goal of the new grant-making program, Exponentiating Mathematics, is to speed up the pace of progress in pure (as opposed to applied) math — and, in doing so, to turn A.I. into a superlative mathematician. 'Mathematics is this great test bed for what is right now the key pain point for A.I. systems,' said Patrick Shafto, a Rutgers University mathematician and computer scientist who now serves as a program manager in DARPA's information innovation office, known as I20. 'So if we overcome that, potentially, it would unleash much more powerful A.I.' He added, 'There's huge potential benefit to the community of mathematicians and to society at large.' Dr. Shafto spoke from his office at DARPA's headquarters, an anonymous building in Northern Virginia whose facade of bluish glass gives little indication that it houses one of the most unusual agencies in the federal government. Inside the building's airy lobby, visitors surrender their cellphones. Near a bank of chairs, a glass display shows a prosthetic arm that can be controlled by the wearer's brain signals. 'By improving mathematics, we're also understanding how A.I. works better,' said Alondra Nelson, who served as a top science adviser in President Joseph R. Biden Jr.'s administration and is a faculty member at the Institute for Advanced Study in Princeton, N.J. 'So I think it's kind of a virtuous cycle of understanding.' She suggested that, down the road, math-adept A.I. could enhance cryptography and aid in space exploration. Started after World War II to compete with the Soviet Union in the space race, DARPA is most famous for fostering the research that led to the creation of ARPANET, the precursor to the internet we use today. At the agency's small gift store, which is not accessible to the public, one can buy replicas of a cocktail napkin on which someone sketched out the rudimentary state of computer networks in 1969. DARPA later funded the research that gave rise to drones and Apple's digital assistant, Siri. But it is also responsible for the development of Agent Orange, the potent defoliant used to devastating effect during the Vietnam War. Want all of The Times? Subscribe.
Yahoo
4 days ago
- Science
- Yahoo
Mathematicians Hunting Prime Numbers Discover Infinite New Pattern for Finding Them
For centuries, prime numbers have captured the imaginations of mathematicians, who continue to search for new patterns that help identify them and the way they're distributed among other numbers. Primes are whole numbers that are greater than 1 and are divisible by only 1 and themselves. The three smallest prime numbers are 2, 3 and 5. It's easy to find out if small numbers are prime—one simply needs to check what numbers can factor them. When mathematicians consider large numbers, however, the task of discerning which ones are prime quickly mushrooms in difficulty. Although it might be practical to check if, say, the numbers 10 or 1,000 have more than two factors, that strategy is unfavorable or even untenable for checking if gigantic numbers are prime or composite. For instance, the largest known prime number, which is 2¹³⁶²⁷⁹⁸⁴¹ − 1, is 41,024,320 digits long. At first, that number may seem mind-bogglingly large. Given that there are infinitely many positive integers of all different sizes, however, this number is minuscule compared with even larger primes. Furthermore, mathematicians want to do more than just tediously attempt to factor numbers one by one to determine if any given integer is prime. 'We're interested in the prime numbers because there are infinitely many of them, but it's very difficult to identify any patterns in them,' says Ken Ono, a mathematician at the University of Virginia. Still, one main goal is to determine how prime numbers are distributed within larger sets of numbers. Recently, Ono and two of his colleagues—William Craig, a mathematician at the U.S. Naval Academy, and Jan-Willem van Ittersum, a mathematician at the University of Cologne in Germany—identified a whole new approach for finding prime numbers. 'We have described infinitely many new kinds of criteria for exactly determining the set of prime numbers, all of which are very different from 'If you can't factor it, it must be prime,'' Ono says. He and his colleagues' paper, published in the Proceedings of the National Academy of Sciences USA, was runner-up for a physical science prize that recognizes scientific excellence and originality. In some sense, the finding offers an infinite number of new definitions for what it means for numbers to be prime, Ono notes. [Sign up for Today in Science, a free daily newsletter] At the heart of the team's strategy is a notion called integer partitions. 'The theory of partitions is very old,' Ono says. It dates back to the 18th-century Swiss mathematician Leonhard Euler, and it has continued to be expanded and refined by mathematicians over time. 'Partitions, at first glance, seem to be the stuff of child's play,' Ono says. 'How many ways can you add up numbers to get other numbers?' For instance, the number 5 has seven partitions: 4 + 1, 3 + 2, 3 + 1 + 1, 2 + 2 + 1, 2 + 1 + 1 + 1 and 1 + 1 + 1 + 1 + 1. Yet the concept turns out to be powerful as a hidden key that unlocks new ways of detecting primes. 'It is remarkable that such a classical combinatorial object—the partition function—can be used to detect primes in this novel way,' says Kathrin Bringmann, a mathematician at the University of Cologne. (Bringmann has worked with Ono and Craig before, and she's currently van Ittersum's postdoctoral adviser, but she wasn't involved with this research.) Ono notes that the idea for this approach originated in a question posed by one of his former students, Robert Schneider, who's now a mathematician at Michigan Technological University. Ono, Craig and van Ittersum proved that prime numbers are the solutions of an infinite number of a particular type of polynomial equation in partition functions. Named Diophantine equations after third-century mathematician Diophantus of Alexandria (and studied long before him), these expressions can have integer solutions or rational ones (meaning they can be written as a fraction). In other words, the finding shows that 'integer partitions detect the primes in infinitely many natural ways,' the researchers wrote in their PNAS paper. George Andrews, a mathematician at Pennsylvania State University, who edited the PNAS paper but wasn't involved with the research, describes the finding as 'something that's brand new' and 'not something that was anticipated,' making it difficult to predict 'where it will lead.' The discovery goes beyond probing the distribution of prime numbers. 'We're actually nailing all the prime numbers on the nose,' Ono says. In this method, you can plug an integer that is 2 or larger into particular equations, and if they are true, then the integer is prime. One such equation is (3n3 − 13n2 + 18n − 8)M1(n) + (12n2 − 120n + 212)M2(n) − 960M3(n) = 0, where M1(n), M2(n) and M3(n) are well-studied partition functions. 'More generally,' for a particular type of partition function, 'we prove that there are infinitely many such prime detecting equations with constant coefficients,' the researchers wrote in their PNAS paper. Put more simply, 'it's almost like our work gives you infinitely many new definitions for prime,' Ono says. 'That's kind of mind-blowing.' The team's findings could lead to many new discoveries, Bringmann notes. 'Beyond its intrinsic mathematical interest, this work may inspire further investigations into the surprising algebraic or analytic properties hidden in combinatorial functions,' she says. In combinatorics—the mathematics of counting—combinatorial functions are used to describe the number of ways that items in sets can be chosen or arranged. 'More broadly, it shows the richness of connections in mathematics,' she adds. 'These kinds of results often stimulate fresh thinking across subfields.' Bringmann suggests some potential ways that mathematicians could build on the research. For instance, they could explore what other types of mathematical structures could be found using partition functions or look for ways that the main result could be expanded to study different types of numbers. 'Are there generalizations of the main result to other sequences, such as composite numbers or values of arithmetic functions?' she asks. 'Ken Ono is, in my opinion, one of the most exciting mathematicians around today,' Andrews says. "This isn't the first time that he has seen into a classic problem and brought really new things to light.' There remains a glut of open questions about prime numbers, many of which are long-standing. Two examples are the twin prime conjecture and Goldbach's conjecture. The twin prime conjecture states that there are infinitely many twin primes—prime numbers that are separated by a value of two. The numbers 5 and 7 are twin primes, as are 11 and 13. Goldbach's conjecture states that 'every even number bigger than 2 is a sum of two primes in at least one way,' Ono says. But no one has proven this conjecture to be true. 'Problems like that have befuddled mathematicians and number theorists for generations, almost throughout the entire history of number theory,' Ono says. Although his team's recent finding doesn't solve those problems, he says, it's a profound example of how mathematicians are pushing boundaries to better understand the mysterious nature of prime numbers.
Malay Mail
7 days ago
- General
- Malay Mail
Letting children write their own stories — Mohd Fadzil Jamaludin
JUNE 14 — Father's Day often invites us to reflect on the men who shaped us. As a 46-year-old researcher, a professional in the world of science and academia, my life has been a tapestry woven with both precision and unpredictability. I come from a large family – 10 siblings, each with our own stories, but all sharing the same foundation: a father who is, in many ways, the archetype of the old Malay tradition, but carries with him a progressive wisdom. He is a man of few words, his presence steady but understated, a quiet force that anchored our sprawling household. He recently celebrated his 71st birthday earlier this month, a milestone that reminds me of the enduring strength behind his calm demeanour. My father was a mathematics professor. As a child, I often watched him draw what seemed to be random scribbles and doodles. Only later did I learn those were integral notations and mathematical curves. I often wondered how he could do complex maths without writing any numbers – just flowing lines and abstract symbols. It was a silent kind of genius, one that didn't need loud explanations or grand declarations. Growing up, I looked up to him with a mix of awe and curiosity. He never mapped out my future or laid down plans for success. Instead, he led by example – his actions speaking volumes where words were sparse. I found myself drawn to academia, perhaps subconsciously following in his purposeful path, believing that I could chart my own course with the same steady resolve. To the author, true fatherhood lies not in directing our children towards conventional achievements, but in nurturing their individual gifts. — Unsplash pic Before marriage, I imagined fatherhood would be a matter of careful planning. I envisioned raising children with the same scientific rigour I applied in my research: structured, logical, and perhaps even predictable. I thought my children would naturally follow in my path, sharing my passion for science and research, just as I had with my father. But life, as it often does, had other plans. Raising five children – some of them now teenagers—has been less a controlled experiment and more a lesson in adaptability. Each day brings new variables, unexpected results, and the humbling realisation that parenting is, above all, a work in progress. My children, each with their own dreams and inclinations, have chosen to pursue the arts rather than the sciences. Their interests diverge from my own, and at times, I struggle to reconcile my expectations with their aspirations. Yet, in these moments of uncertainty, I find myself returning to my father's example. He never imposed his will on me; he allowed me the freedom to discover my own path. His quiet support, his unwavering presence, taught me that fatherhood is not about moulding our children in our own image, but about giving them the space to become who they are meant to be. This Father's Day, I celebrate my father – not for the plans he made, but for the plans he allowed me to make for myself. In our Malaysian society, where academic and professional success often define parental pride, his quiet wisdom reminds me that true fatherhood lies not in directing our children towards conventional achievements, but in nurturing their individual gifts. As I watch my children pursue the arts, humanities, and creative fields in a culture that traditionally prizes the sciences, I'm learning that the greatest 'kejayaan' we can offer them is the confidence to define success on their own terms. To all Malaysian fathers learning to celebrate their children' unique journeys – Happy Father's Day. *Mohd Fadzil Jamaludin is a research officer at the Faculty of Engineering, Universiti Malaya, and may be reached at [email protected] ** This is the personal opinion of the writer or publication and does not necessarily represent the views of Malay Mail.
Geeky Gadgets
13-06-2025
- Science
- Geeky Gadgets
Mathematicians Stunned as OpenAI o3-mini Answers World's Hardest Math Problems
What if the secrets to the universe's most perplexing mathematical riddles were no longer locked away, but instead cracked open by an artificial mind? In a new development, OpenAI's o3-mini model has achieved what many thought impossible: solving problems that have baffled mathematicians for centuries. From the intricate depths of number theory to the abstract landscapes of topology, this AI-driven system has ventured where human intuition alone has struggled to tread. But this isn't just a story about equations and algorithms—it's a profound moment of reckoning for the relationship between human ingenuity and machine intelligence. Wes Roth provides more insights into how the o3-mini is reshaping the mathematical frontier, offering not just solutions but entirely new ways of thinking. You'll uncover how this AI model identifies hidden patterns in complex datasets, proposes new approaches to unsolved problems, and even collaborates with researchers to verify proofs. Yet, as with any disruptive innovation, its rise sparks debate: Will AI complement human creativity or overshadow it? By the end, you may find yourself questioning not just the future of mathematics, but the evolving nature of discovery itself. AI Solves Complex Math What Sets OpenAI o3-mini Apart? The o3-mini model, developed by OpenAI, represents a new advancement in AI-driven problem-solving. Unlike traditional computational tools that rely on predefined algorithms, o3-mini uses machine learning to analyze patterns, uncover solutions, and propose innovative approaches to highly complex problems. For instance, the model has successfully tackled unresolved equations in number theory and topology—fields renowned for their intricate and abstract nature. These achievements highlight o3-mini's ability to process vast datasets, identify hidden relationships, and generate insights that were previously unattainable. By doing so, it demonstrates the potential for AI to complement human expertise in solving problems once deemed insurmountable. This capability positions o3-mini as a unique tool in the evolving landscape of mathematical research. Transforming Mathematical Research with AI Artificial intelligence is rapidly becoming an essential tool in mathematical research, offering capabilities that extend far beyond simple automation. By handling repetitive calculations and exploring vast solution spaces, AI allows researchers to dedicate more time to higher-level conceptual thinking and creative problem-solving. The o3-mini model exemplifies this transformation. Beyond solving problems, it provides detailed explanations of its reasoning, making sure its findings are both accessible and transparent. This feature is particularly valuable in fields such as algebraic geometry and combinatorics, where innovative approaches are often required to address intricate challenges. Moreover, AI systems like o3-mini can assist in verifying proofs, making sure accuracy, and streamlining the peer-review process, which significantly reduces the time and effort required by researchers. In addition to these practical applications, AI is also reshaping the way researchers collaborate. By offering new tools and perspectives, models like o3-mini encourage interdisciplinary approaches, fostering innovation across diverse areas of mathematics and beyond. OpenAI o3-mini Solving Math Problems Watch this video on YouTube. Here is a selection of other guides from our extensive library of content you may find of interest on OpenAI o3-mini. Addressing Unsolved Mathematical Mysteries One of the most compelling aspects of o3-mini's capabilities is its potential to tackle long-standing mathematical mysteries. Problems such as the Riemann Hypothesis or the Birch and Swinnerton-Dyer Conjecture, which have eluded mathematicians for centuries, may now be closer to resolution. By analyzing historical data, testing hypotheses, and generating new lines of inquiry, AI models like o3-mini provide fresh perspectives on these enduring puzzles. While it remains uncertain whether AI will definitively solve such problems, the progress demonstrated by o3-mini suggests that these possibilities are no longer purely speculative. This development represents a significant step toward a future where AI plays a central role in advancing mathematical understanding and addressing some of the most profound questions in the field. Reactions from the Mathematical Community The achievements of o3-mini have elicited a range of reactions from the mathematical community, reflecting both excitement and caution. On one hand, many researchers are optimistic about AI's potential to accelerate discovery and innovation. They see tools like o3-mini as valuable collaborators that can enhance human creativity and expand the boundaries of what is possible in mathematical research. On the other hand, some express concerns about the implications of relying too heavily on AI. Critics worry that the increasing use of AI could diminish the role of human intuition and creativity, which have traditionally been central to mathematical breakthroughs. They argue that mathematics is not just about finding solutions but also about the process of discovery and the insights gained along the way. Proponents of AI counter these concerns by emphasizing the collaborative potential of these technologies. Rather than replacing mathematicians, AI can serve as a powerful partner, augmenting human capabilities and opening new avenues for exploration. This perspective highlights the importance of maintaining a balance between using AI's strengths and preserving the uniquely human aspects of mathematical inquiry. The Future of Mathematics and Artificial Intelligence The success of OpenAI's o3-mini model marks a pivotal moment in the evolving relationship between artificial intelligence and mathematics. As AI systems continue to advance, their applications in mathematical research are expected to expand, influencing education, industry, and academia in profound ways. For those interested in the intersection of technology and mathematics, this development offers a glimpse into a future where AI not only solves problems but also inspires new ways of thinking about them. Whether it's advancing theoretical research, addressing practical challenges, or fostering interdisciplinary collaboration, AI is poised to play an increasingly central role in shaping the mathematical landscape. This progress also raises important questions about how AI will influence the next generation of mathematicians. As AI tools become more integrated into research and education, they may redefine how mathematical concepts are taught and understood, encouraging students and researchers alike to approach problems from new perspectives. The collaboration between human ingenuity and machine intelligence is unlocking new frontiers, offering exciting possibilities for the future of mathematics. Media Credit: Wes Roth Filed Under: AI, Top News Latest Geeky Gadgets Deals Disclosure: Some of our articles include affiliate links. If you buy something through one of these links, Geeky Gadgets may earn an affiliate commission. Learn about our Disclosure Policy.
