Latest news with #mathematicians
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.
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.
Daily Mail
5 days ago
- General
- Daily Mail
Simple looking math problem leaves people stumped - are you smarter than a 5th grader?
A simple arithmetic problem has left internet users scratching their heads - with only a few landing on the correct solution to this elementary school-level equation. While most people learned the concept needed to simplify these kinds of equations by the time they turned 10, many attempters drew a blank this time around. The key to getting to the bottom of this seemingly straightforward problem is to call upon every mathematicians' favorite acronym - PEMDAS. Each letter represents a math symbol in the order they are meant to be done when they come up in a problem. Anything inside parentheses (P) should be worked out first. Secondly, exponents (E) should be addressed. Next should be multiplication and division (MD), but multiplication does not necessarily have to come before division. The correct method is to address them from left to right as the are written in the equation. Lastly, addition and subtraction (AS). When only those two operations remain, the sum can be solved from left to right because the order makes no difference. Here's the equation, give it a go before reading on: Under the X post, shared by user BreakTheSilos, that first sparked the mathematical confusion, some users confidently claimed the answer was 0 or 1. But both solutions are incredibly incorrect. Applying PEMDAS, the first step to solving this problem is to tackle what's inside the parentheses: 3 + 5. After performing basic addition, you get 5, making the new equation 45 ÷ 9 (5). At this stage, some people fell into a trap. Instead of following PEMDAS, they decided to multiply 9 and 5, leaving them with the flawed expression 45 ÷ 45. Instead of this faulty method, you must remember that multiplication and division is to address them from left to right. Keeping that in mind, the correct second step is to solve 45 ÷ 9, which equals 5. Now, left with 5(5), the only thing to do is to multiply 5 x 5, finally getting you the correct answer of 25.
The Herald
12-05-2025
- Business
- The Herald
Lessons from Armenia to engineer a better future for youth
Armenia is a West Asian country that is a bit like Lesotho. Like Lesotho, it is landlocked and not super-endowed with natural minerals. It has a population of 2.7-million people, while Lesotho has about 2.2-million. Like Lesotho's working people, most Armenians leave their home country to seek jobs elsewhere. Armenia has done something spectacular: it teaches its children mathematics. Before the collapse of communism 35 years ago, it was one of the mathematics hubs of the Union of Soviet Socialist Republics (USSR). It continued the traditions in the post-Soviet era and is known for 'exporting' mathematicians, engineers, scientists, programmers, coders and other tech boffins across the globe. Now, listen to this carefully: The BBC reported recently that, 11 years ago, Armenia launched a school programme, in partnership with the private sector, called 'Armath' (which means 'root' in English). It teaches children programming, robotics, coding, 3D modelling and other subjects. There are now 650 Armath laboratories in schools across Armenia, with more than 600 teachers and 17,000 students writing code and inventing apps and gadgets of all kinds. Armath started because the country wanted 'to see Armenia becoming a tech centre powerhouse that delivers utmost values to Armenia and to the world'. And in just 11 years it is headed there: Armenia, with just 2.7-million people, has 4,000 tech firms (SA has 650). 'One floated in New York in December 2024, and is now worth more than $10bn (R182.1bn),' the BBC article said. What does this tell us? A country needs a vision, a strategy and the political will to see that strategy through. Armenia had that will, hence the success of the programme. Last week, I came across a story that broke my heart. We come across so many of these heartbreaking stories in SA and the world these days that we tend to ignore them. They have become normalised. I came across one such story last Thursday, and it reminded me that we must not stop raging against such betrayal of SA's poorest and most vulnerable. The story is particularly cutting because it is an assault on our children, the people who will inherit this place. It showed how we are robbing them of a future. Depressingly, there is no outrage, no noise, no shock or horror at this story. The ministry of education announced last week that 464 public schools do not offer mathematics to their pupils. Yes, 464 schools do not offer maths to pupils. Of these schools, 135 are in KwaZulu-Natal, 84 in the Eastern Cape, 78 in Limpopo and 61 are in the Western Cape. The rest are in Gauteng and the North West (31 schools each), the Northern Cape with 19, the Free State with 14 and Mpumalanga with 11. It's not as if this is a new problem: the percentage of pupils opting for maths declined from 46% in 2011 to 34% in 2023. In 2024, only 255,762 pupils registered for the subject, down from 268,100 in 2023. The education ministry said 'schools may not have sufficient resources or demand to offer both mathematics and mathematical literacy'. Let us be perfectly clear and honest with each other here. This is not about lack of teachers. This is about leadership. Our political 'leaders' have for 31 years not cared about the education of our children and still do not care today. If they cared, then education, and maths education in particular, would have been top of their agenda. As these statistics show, they do not care. Over the past 30 years, our political leaders sent their children to 'formerly white' schools and abandoned township and rural schools. Local councillors, teachers, principals, packed their children into dangerous minibus taxis to schools in the formerly white suburbs. Township and rural schools were left to rot without teachers to lead pupils in maths or science. This lack of leadership, this lack of a plan to make every school in rural areas and in townships a centre of excellence, is where we failed. This is a spectacular failure of vision, strategy and leadership. Many of our politicians run around telling poor people to kick out foreigners who run shops in townships. What these politicians don't say is that we have stripped many of our own children of the ability to run their own spaza shops. They can't count, add, multiply or divide, let alone determine a profit margin on a packet of sweets. We tell our children to be entrepreneurs, yet we strip them of the ability to even start. That this is not a national emergency underlines just how awfully inadequate and ill-equipped for leadership our politicians are. We are faced with a disaster here. This is how countries collapse. This is why teeny-weeny countries like Armenia manage to wipe the floor with us in every way. We are led by people who have no clue what it takes to build a successful, prosperous and durable country.
