Latest news with #primeNumbers
Sustainability Times
03-07-2025
- Science
- Sustainability Times
'Prime Numbers Had a Hidden Code': Mathematician Cracks 2,000-Year-Old Mystery That Could Rewrite Number Theory
IN A NUTSHELL 🔍 Mathematician Ken Ono discovered a surprising link between prime numbers and integer partitions , reshaping our understanding of these elusive integers. and , reshaping our understanding of these elusive integers. 🛡️ Prime numbers play a crucial role in modern cryptography , underpinning secure communications and transactions through their inherent complexity. , underpinning secure communications and transactions through their inherent complexity. 🔗 The discovery connects two distinct mathematical fields, bridging the gap between combinatorics and number theory with innovative equations. and with innovative equations. 🔮 This breakthrough opens new research avenues, prompting questions about its potential applications to other numerical structures and the future of mathematical exploration. The world of numbers has often been a realm of mysteries and discoveries, and nothing epitomizes this better than prime numbers. These elusive integers, only divisible by themselves and one, appear randomly along the number line, defying prediction and order. Yet, a recent breakthrough may change our perspective on these fundamental components of arithmetic. Mathematician Ken Ono and his team have uncovered an unsuspected link between prime numbers and a completely different mathematical field: integer partitions. This connection could revolutionize our understanding of prime numbers and unveil a hidden pattern in what was once considered pure randomness. The Ancient Quest for Primes: Revisiting the Sieve To appreciate the significance of this breakthrough, we must journey back to the third century BCE. It was then that the Greek scholar Eratosthenes devised an elegantly simple method to identify prime numbers—known today as the 'Sieve of Eratosthenes.' This technique involves systematically eliminating the multiples of each integer, leaving only those that remain indomitable: the primes. Despite its antiquity, the sieve remains one of the most effective tools for sifting through these unique integers. This enduring relevance underscores the complexity of the problem at hand: even after more than 2,000 years of research, no straightforward algorithm or universal formula can predict where the next prime number will appear. This ancient method highlights the persistent challenge prime numbers pose. While it is a rudimentary yet powerful tool, the quest to fully comprehend primes continues, emphasizing their profound mystery and significance in mathematics. 'Like a Floating Magic Carpet': Newly Discovered Deep-Sea Creature Stuns Scientists With Its Surreal, Otherworldly Movements Why Prime Numbers Matter Today Beyond their theoretical allure, prime numbers hold immense practical importance in our modern lives. Every time you send an encrypted message, complete a secure transaction, or connect to a website via HTTPS, you rely—perhaps unknowingly—on their power. Modern cryptography, particularly the RSA system, is based on the difficulty of factoring large prime numbers. This complexity is crucial for cybersecurity, yet it also makes primes frustratingly elusive for mathematicians. The difficulty in factoring these numbers ensures the security of our digital communications, highlighting the dual nature of primes as both a challenge and a protector in the digital age. The paradox of prime numbers lies in their dual role: they are both a foundational mathematical enigma and a critical component of our digital security infrastructure. 'Time Breaks Down at Quantum Scale': New Scientific Discovery Shocks Physicists and Redefines the Laws of the Universe An Unexpected Connection: Prime Numbers and Integer Partitions Here is where the story takes an unexpected turn. Ken Ono and his team have found that prime numbers are not as chaotic as once believed. In fact, they can be detected through an infinite number of ways, using equations derived from a seemingly unrelated mathematical object: the integer partition function. But what exactly is an integer partition? It is a way of breaking down a whole number into the sum of positive integers. For instance, the number 4 can be expressed in several ways: 4 3 + 1 2 + 2 2 + 1 + 1 1 + 1 + 1 + 1 Though simple in appearance, integer partitions conceal immense combinatorial complexity. These partitions are at the heart of the discovery. Researchers have shown that prime numbers can be identified as solutions to an infinite number of Diophantine equations, crafted from partition functions. This discovery not only bridges two previously distinct areas of mathematics but also opens new avenues for exploration. 'Google Just Changed Everything': This Ruthless New AI Reads 1 Million Human DNA Letters Instantly and Scientists Are Stunned A Breakthrough Celebrated by the Mathematical Community This groundbreaking discovery has been hailed by the mathematical community as 'remarkable.' Professor Kathrin Bringmann from the University of Cologne, an expert in the field, emphasizes the newfound capability of the partition function to detect prime numbers, opening entirely new fields of inquiry. In essence, this breakthrough is not just a theoretical accomplishment; it connects two previously distant mathematical territories, creating an unexpected bridge between combinatorics and number theory. This discovery is a testament to the evolving nature of mathematics, where long-studied concepts can yield new insights and cross-disciplinary connections. As we delve into the mysteries of prime numbers, new questions arise. Can this approach be used to gain insights into other numerical structures? Are there equivalents for composite numbers, arithmetic sequences, or other enigmatic objects? As is often the case in mathematics, each discovery opens a multitude of new chapters to explore. With quantum computing on the horizon, redefining our theoretical foundations is not merely an academic pursuit—it is a strategic necessity. Could this be the beginning of a new era in our understanding of numbers? Our author used artificial intelligence to enhance this article. Did you like it? 4.5/5 (27)
South China Morning Post
28-06-2025
- Science
- South China Morning Post
Trailblazing mathematician Yitang Zhang leaves US for job at Chinese university
Chinese-American mathematician Yitang Zhang has left the United States to join Sun Yat-sen University in southern China as a full-time professor. The 70-year-old number theorist has been appointed to the university's newly established Institute of Advanced Study Hong Kong and will live and work in the Greater Bay Area , the university announced at a ceremony on Friday afternoon. While the university did not give further details about his appointment, it noted that Zhang had relocated to China with his family. Zhang spent a decade as a mathematics professor at the University of California, Santa Barbara. Zhang is best known for his groundbreaking work on prime numbers. In a 2013 paper published in the Annals of Mathematics, he proved for the first time that infinitely many pairs of prime numbers are separated by a bounded gap. The result marked a major step towards solving the twin prime conjecture – a fundamental unsolved problem in mathematics – and reignited global interest in number theory. Born in Shanghai in 1955, Zhang showed a strong talent for maths from an early age. He was unable to attend high school during the Cultural Revolution but taught himself and was admitted to Peking University in 1978. In 1985, he went to Purdue University in the US for his PhD. After graduating in 1991, he struggled to find an academic position. To make ends meet, Zhang worked various jobs, including as an accountant, restaurant manager, and Chinese food delivery driver, and at one point lived in his car.
Yahoo
20-06-2025
- 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.
Yahoo
18-06-2025
- Science
- Yahoo
A Mathematician Found a Hidden Pattern That Could Keep Your Biggest Secrets Safe
Here's what you'll learn when you read this story: Prime numbers are essential for technologies like RSA encryption, which rely on the difficulty of guessing these numerals. A new paper shows that another area of mathematics called integer partition unlocks 'infinitely many new ways' to detect them beyond divisibility. The team said that this breakthrough arrived by using decades-old methods to answer mathematical questions no one else thought to ask. Although prime numbers are a mathematical concept everyone learns about in elementary school, extremely large prime numbers form the backbone of some of the most complex technologies in modern society—especially in the realm of cryptography. But in the burgeoning era of quantum computers, which can solve problems exponentially faster than standard computers (including supercomputers), there's a chance that this kind of previously uncrackable protection could suddenly become very vulnerable. This has pushed mathematicians—including Ken Ono from the University of Virginia—to continue exploring the frontiers of prime numbers. In September of last year, Ono (along with co-authors William Craig and Jan-Willem van Ittersum) published a paper in the journal Proceedings of the National Academy of Sciences (PNAS) exploring how to find new prime numbers with a novel approach centered around what are called integer partitions. His groundbreaking work scored him recognition for the Cozzarelli Award for originality and creativity, but to understand it, we'll need to take a few steps back. A prime number (as you likely know) is an integer that is not divisible by any number other than 1 and itself. While there are technically infinite prime numbers, it's difficult to find new ones, as they appear in a number line with no pattern. (Currently, the largest known prime number is more than 41 million digits long.) But Ono and his co-authors discovered a connection between prime numbers and integer partitions, which divvy up numbers into all their possible smaller sums—the number four, for example, can be described as 4, as 3 + 1, as 2 + 2, as 2 + 1 + 1, and as 1 + 1 + 1 + 1. 'The prime numbers, the building blocks of multiplicative number theory, are the solutions of infinitely many special 'Diophantine equations' in well-studied partition functions,' the authors wrote. 'In other words, integer partitions detect the primes in infinitely many natural ways.' Named for the third-century mathematician Diophantus of Alexandria, these equations can be incredibly complex, but if the resulting answer turns out to be true, that means you're working with a prime number. This essentially devises a new way to investigate prime numbers that has never been explored before. 'This paper, as excited as I am about it, represents theoretical math that could've been done decades ago,' Ono said in a video interview accompanying a press statement. 'What I like about our theorem is that if there was a time machine, I could go back to 1950, explain what we done, and it would generate the same level of excitement […] and the experts at that time would understand what we did.' Ono is intimately familiar with the security implications of prime number research, as he serves on the advisory board for the National Security Agency (NSA). Technologies like RSA encryption rely on the difficulty of detecting prime numbers to safeguard the world's most sensitive information, so understanding prime numbers from every conceivable angle will be helpful when quantum computers make ferreting out these unfathomably large numbers easier. Speaking with Scientific American, many mathematicians say this work serves as the foundation of a new way of seeing what other mathematical connections can be made using partition functions. Prime numbers may be elementary, but they remain a fixture of our complex technological future. You Might Also Like The Do's and Don'ts of Using Painter's Tape The Best Portable BBQ Grills for Cooking Anywhere Can a Smart Watch Prolong Your Life?
Yahoo
16-06-2025
- 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.
