All articles tagged with #prime numbers
Anthropic's AI model Claude failed to solve the Riemann hypothesis, but its partial work inspired mathematician Youness Lamzouri to make a breakthrough on a related prime number problem.
Two vast broods of periodical cicadas—Brood XIX (13-year) and Brood XIII (17-year)—emerged simultaneously across the eastern U.S. in 2024, the first such event since 1803, with next co-emergence not until 2245. The 13- and 17-year cycles are prime numbers, a fitness strategy explained as reducing overlap with predators’ shorter life cycles, a concept supported by predator-avoidance theory and, in part, by hybridization hypotheses. In mass emergences, predator satiation further boosts cicada survival, making these prime cycles a striking example of how a mathematical property can influence evolution.}
The article discusses the significance of prime numbers in mathematics, highlighting the biggest unsolved problem, the Riemann Hypothesis, which concerns the distribution of primes and has profound implications for understanding their behavior. Despite centuries of study, many prime-related questions remain open, but recent progress and optimism suggest that solutions, especially for the Riemann Hypothesis, may be within reach, potentially revolutionizing mathematics.
Mathematicians have discovered a remarkable new connection between prime numbers and partition functions, revealing infinitely many ways to detect primes without divisibility checks, which could influence future research in number theory and cryptography.
Two mathematicians, Ben Green and Mehtaab Sawhney, have developed a new method using additive combinatorics and Gowers norms to study the distribution of prime numbers, successfully proving that infinitely many primes fit the form p² + 4q², which opens new avenues for understanding prime patterns and their applications in fields like cryptography.
Researchers at City University of Hong Kong and North Carolina State University have made a groundbreaking discovery in prime number theory, challenging the long-held belief that prime numbers are unpredictable. Their research has led to the development of a Periodic Table of Primes (PTP) that can accurately predict the locations of prime numbers, with potential applications in fields such as cybersecurity. This breakthrough has the potential to significantly impact encryption and cryptography, making data more secure.
The largest known prime number is 2^(82,589,933) - 1, also known as M82589933, which has a staggering 24,862,048 digits. It was discovered in 2018 by the Great Internet Mersenne Prime Search (GIMPS), a distributed computing project that involves volunteers running software on their computers. Mersenne primes, named after the French monk Marin Mersenne, are a specific type of prime number. GIMPS has found a total of 17 Mersenne primes, with M82589933 being the latest discovery. Finding larger prime numbers continues to be a quest for mathematicians, and while there are strategies for determining if Mersenne numbers are prime, the search for the next one remains unpredictable.
High school student Daniel Larsen has proven a new theorem about Carmichael numbers, a type of not-quite-prime number, that had eluded mathematicians for decades. This theorem is related to Pierre de Fermat's "little theorem," which states that for any prime number, the quantity a^p - a is divisible by p for any integer a. While the converse of Fermat's little theorem is not true, Larsen's work explores the distribution of Carmichael numbers and builds on the research of famous mathematicians such as James Maynard and Terence Tao.