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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.
Gizmodo
31-05-2025
- General
- Gizmodo
A Brief History of Our Obsession With Prime Numbers—and Where the Hunt Goes Next
A shard of smooth bone etched with irregular marks dating back 20,000 years puzzled archaeologists until they noticed something unique – the etchings, lines like tally marks, may have represented prime numbers. Similarly, a clay tablet from 1800 B.C.E. inscribed with Babylonian numbers describes a number system built on prime numbers. As the Ishango bone, the Plimpton 322 tablet and other artifacts throughout history display, prime numbers have fascinated and captivated people throughout history. Today, prime numbers and their properties are studied in number theory, a branch of mathematics and active area of research today. A history of prime numbers Informally, a positive counting number larger than one is prime if that number of dots can be arranged only into a rectangular array with one column or one row. For example, 11 is a prime number since 11 dots form only rectangular arrays of sizes 1 by 11 and 11 by 1. Conversely, 12 is not prime since you can use 12 dots to make an array of 3 by 4 dots, with multiple rows and multiple columns. Math textbooks define a prime number as a whole number greater than one whose only positive divisors are only 1 and itself. Math historian Peter S. Rudman suggests that Greek mathematicians were likely the first to understand the concept of prime numbers, around 500 B.C.E. Around 300 B.C.E., the Greek mathematician and logician Euler proved that there are infinitely many prime numbers. Euler began by assuming that there is a finite number of primes. Then he came up with a prime that was not on the original list to create a contradiction. Since a fundamental principle of mathematics is being logically consistent with no contradictions, Euler then concluded that his original assumption must be false. So, there are infinitely many primes. The argument established the existence of infinitely many primes, however it was not particularly constructive. Euler had no efficient method to list all the primes in an ascending list. In the middle ages, Arab mathematicians advanced the Greeks' theory of prime numbers, referred to as hasam numbers during this time. The Persian mathematician Kamal al-Din al-Farisi formulated the fundamental theorem of arithmetic, which states that any positive integer larger than one can be expressed uniquely as a product of primes. From this view, prime numbers are the basic building blocks for constructing any positive whole number using multiplication – akin to atoms combining to make molecules in chemistry. Prime numbers can be sorted into different types. In 1202, Leonardo Fibonacci introduced in his book 'Liber Abaci: Book of Calculation' prime numbers of the form (2p – 1) where p is also prime. Today, primes in this form are called Mersenne primes after the French monk Marin Mersenne. Many of the largest known primes follow this format. Several early mathematicians believed that a number of the form (2p – 1) is prime whenever p is prime. But in 1536, mathematician Hudalricus Regius noticed that 11 is prime but not (211 – 1), which equals 2047. The number 2047 can be expressed as 11 times 89, disproving the conjecture. While not always true, number theorists realized that the (2p – 1) shortcut often produces primes and gives a systematic way to search for large primes. The search for large primes The number (2p – 1) is much larger relative to the value of p and provides opportunities to identify large primes. When the number (2p – 1) becomes sufficiently large, it is much harder to check whether (2p – 1) is prime – that is, if (2p – 1) dots can be arranged only into a rectangular array with one column or one row. Fortunately, Édouard Lucas developed a prime number test in 1878, later proved by Derrick Henry Lehmer in 1930. Their work resulted in an efficient algorithm for evaluating potential Mersenne primes. Using this algorithm with hand computations on paper, Lucas showed in 1876 that the 39-digit number (2127 – 1) equals 170,141,183,460,469,231,731,687,303,715,884,105,727, and that value is prime. Also known as M127, this number remains the largest prime verified by hand computations. It held the record for largest known prime for 75 years. Researchers began using computers in the 1950s, and the pace of discovering new large primes increased. In 1952, Raphael M. Robinson identified five new Mersenne primes using a Standard Western Automatic Computer to carry out the Lucas-Lehmer prime number tests. As computers improved, the list of Mersenne primes grew, especially with the Cray supercomputer's arrival in 1964. Although there are infinitely many primes, researchers are unsure how many fit the type (2p – 1) and are Mersenne primes. By the early 1980s, researchers had accumulated enough data to confidently believe that infinitely many Mersenne primes exist. They could even guess how often these prime numbers appear, on average. Mathematicians have not found proof so far, but new data continues to support these guesses. George Woltman, a computer scientist, founded the Great Internet Mersenne Prime Search, or GIMPS, in 1996. Through this collaborative program, anyone can download freely available software from the GIMPS website to search for Mersenne prime numbers on their personal computers. The website contains specific instructions on how to participate. GIMPS has now identified 18 Mersenne primes, primarily on personal computers using Intel chips. The program averages a new discovery about every one to two years. The largest known prime Luke Durant, a retired programmer, discovered the current record for the largest known prime, (2136,279,841 – 1), in October 2024. Referred to as M136279841, this 41,024,320-digit number was the 52nd Mersenne prime identified and was found by running GIMPS on a publicly available cloud-based computing network. This network used Nvidia chips and ran across 17 countries and 24 data centers. These advanced chips provide faster computing by handling thousands of calculations simultaneously. The result is shorter run times for algorithms such as prime number testing. The Electronic Frontier Foundation is a civil liberty group that offers cash prizes for identifying large primes. It awarded prizes in 2000 and 2009 for the first verified 1 million-digit and 10 million-digit prime numbers. Large prime number enthusiasts' next two challenges are to identify the first 100 million-digit and 1 billion-digit primes. EFF prizes of US$150,000 and $250,000, respectively, await the first successful individual or group. Eight of the 10 largest known prime numbers are Mersenne primes, so GIMPS and cloud computing are poised to play a prominent role in the search for record-breaking large prime numbers. Large prime numbers have a vital role in many encryption methods in cybersecurity, so every internet user stands to benefit from the search for large prime numbers. These searches help keep digital communications and sensitive information safe. Jeremiah Bartz, Associate Professor of Mathematics, University of North Dakota. This article is republished from The Conversation under a Creative Commons license. Read the original article.
The Guardian
26-05-2025
- General
- The Guardian
Did you solve it? The most Guardian puzzle ever
Earlier today I set you some problems on 'guardian numbers.' Here they are again with solutions. The definition: the guardian of x is the next number that shares at least one digit of x. For example: the guardian of 4 is 14, and the guardian of 59 is 65. Guardian figures (For all the questions, consider only the numbers 1, 2, 3, 4, and so on.) a) Find the guardian of 17, the guardian of 79, and the guardian of 179. b) Find two consecutive numbers with the same guardian. c) How many numbers are exactly 9 less than their guardian? Solutions a) 21, 87, 180. b) 19, 20, also 199, 200, and so on. c) one (the number 1) A number's grandguardian is their guardian's guardian. d) Who is the grandguardian of 499? e) Whose grandguardian is 900? Solutions d) 505, e) 898 Two numbers are cousins if they have the same grandguardian but different guardians. f) What is the smallest pair of cousins? Solution f) 28, 30 Thanks again to Daniel Griller, who set today's puzzle and came up with the idea of number guardians. Check out his book A Ring of Cats and Dogs. I've been setting a puzzle here on alternate Mondays since 2015. I'm always on the look-out for great puzzles. If you would like to suggest one, email me.
The Guardian
26-05-2025
- Entertainment
- The Guardian
Can you solve it? The most Guardian puzzle ever
Numbers can be odd, even, prime, square, natural, perfect, complex, rational…and as from today they can also be guardians. Let the numbers be 1, 2, 3, 4, and so on. The guardian of x is the next number that shares at least one digit of x. For example: the guardian of 4 is 14, and the guardian of 59 is 65. Guardian figures a) Find the guardian of 17, the guardian of 79, and the guardian of 179. b) Find two consecutive numbers with the same guardian. c) How many numbers are exactly 9 less than their guardian? A number's grandguardian is their guardian's guardian. d) Who is the grandguardian of 499? e) Whose grandguardian is 900? Two numbers are cousins if they have the same grandguardian but different guardians. f) What is the smallest pair of cousins? I'll be back with the solutions at 5pm. Please NO SPOILERS. Instead please share your favourite type of number or invent your own. The idea of guardian numbers is due to the brilliant puzzle setter Daniel Griller. His most recent book is A Ring of Cats and Dogs. I've been setting a puzzle here on alternate Mondays since 2015. I'm always on the look-out for great puzzles. If you would like to suggest one, email me.
