Prime numbers have fascinated mathematicians for centuries because of their unpredictable and seemingly random distribution. In a groundbreaking preprint study, researchers have devised a new method to enhance our search for ambiguous values, but also revealed the limits of our ability to detect them. .
A prime number is only divisible by 1 and itself. These act as “atoms” in mathematics, allowing other numbers to be broken down into factors (such as 12 = 2 × 2 × 3). As the number increases, it becomes increasingly difficult to identify prime numbers. Where do you start when asked, “How many prime numbers are there between 1 and 1,000?”
The classic Sieve of Eratosthenes provides a starting point. This ancient technique systematically removes the multiples of each prime number, leaving only the prime itself “dropped out.” Mathematicians call the eliminated multiples “Type I information,” and they help us predict how many prime numbers there are in a certain range. However, this information is limited. “Sometimes we get as good Type I information as we could hope for, but we still don’t find a prime number,” explains study co-author Kevin Ford, a mathematician at the University of Illinois at Urbana-Champaign.
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Ford and Oxford University mathematician James Maynard provided a powerful method for studying a wide range of prime numbers by accurately estimating the number of primes that should exist within a prime number. This study combines two complementary perspectives. One is Type I information for numbers that are crossed out (such as crossing out all multiples of 2, then 3), and an explanation for numbers that are crossed out multiple times (such as how 6 appears in a list). multiples of both 2 and 3)—referred to as Type II information.
Mathematicians can adjust how each type of information is weighted to obtain the most accurate number of prime numbers possible within a certain range. But in carefully adjusting these two knobs, the authors discovered a fundamental limitation. It was a precise mathematical boundary beyond which no further adjustment could improve the accuracy of the count, and it revealed a deep truth about how these numbers were distributed on the number line.
In this study, we liken the precision, or “information strength”, of these estimates for a set to changing the size of the mesh of a sieve. If it is too small, it will capture all the numbers. If it is too large, prime numbers will slip through. Kaisa Matomaki, a mathematician who studies the distribution of prime numbers at the University of Turku in Finland, said the study “answers precisely and directly what information is ‘good enough’ to detect prime numbers.” speak To develop a complete theory of prime numbers, it’s important to understand the limitations when designing sieves, adds Princeton University mathematician Peter Cernak, an expert on prime sieve theory. “It is fundamental to clarify what cannot be achieved.”
Ford hopes this technique will help researchers tackle long-standing unsolved questions. “Prime numbers are distributed in a very, very mysterious way, and we’re trying to make a little bit more sense of it.”
