Today, thousands of computers around the world are scouring the number line in a search for rare mathematical gems. Prime number enthusiasts, seeking ever-larger numbers that are divisible by 1 and only themselves, muster vast amounts of computing power and algorithmic ingenuity in an attempt to etch their name into the mathematical history scrolls.
The latest entry is from Luke Durant, a researcher in San Jose, California. Durant’s discovery overturned the former record holder, who had been undisputed for nearly six years, an unprecedentedly long reign in the modern pursuit. increasingly large prime numbers. The gap makes sense: as the former grow, they spread farther, and each new discovery is more difficult than the last.
The new Prime has a staggering 41,024,320 figures. To put that into perspective, the calculated number of atoms it has about 80 digits in the observable universe. Each additional digit increases a number by 10 times, so the magnitude of the new prime lives beyond human comprehension. In pure mathematics, primes play a big roleThey are the main characters of an area called where number theoryand in practice, where, for example, they are widely used on the basis encryption algorithms. A prime with 41 million digits will not be entered immediately useful numbers, but for now, it adds a feather to a community that wants to capture the colossal.
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Durant’s success comes in part from new smart software Excellent Internet Mersenne Prime Search and partly from the heavy hardware and computational muscle he personally mobilized behind them. By assembling a “cloud supercomputer” spanning 17 countries, he ended a long tradition of personal computer firsts.
Prime numbers are often called the “building blocks of mathematics” because every integer greater than 1 has a fingerprint as the product of a single collection of primes. For example, 15 is the product of the prime numbers 5 and 3, 13 cannot be divided like this because 13 is prime. The study of these numbers dates back at least to the ancient Greeks. In the year 300 BC Euclid proven in his textbook The elements that infinite prime numbers existand mathematicians, both professional and amateur, have enjoyed their hunt ever since.
While finding the first string—2, 3, 5, 7, 11, and so on—is easy, the task becomes considerably more difficult as the numbers increase. For millennia, people discovered the first ones by hand—1951. until the year when computers took over the search. But even silicon bounty hunters have difficulty spotting primes far down the number line, as testing the preference of a large number is not trivial. To cope, researchers use every little optimization trick they can to speed up testing or reduce the hunting ground, thereby increasing their chances of finding a new first.
Consider the number 99,400,891. How would you determine if it is the first? Simply divide by each smaller number and check if it has a divisor (besides 1 and so). But that’s nearly 100 million cases to verify a relatively small eight-figure number. You’d save significant work by realizing that you don’t just have to check all the numbers all the way to the target the first the numbers Why? You need to find a unique divisor (a number that divides 99,400,891 cleanly without a remainder). We know that any non-prime divisor can be further divided into its prime factors; if your target is divisible by 15, then it is also divisible by the prime numbers 5 and 3, so you must check the latter to determine priority. A further saving would be that you don’t have to check all the smaller primes, only the ones up to the square root of 99,400,891 (the number multiplied by itself that gives you this eight-digit result). If one of those smaller primes doesn’t divide cleanly, then you can stop looking because it will exceed the product of two numbers greater than the square root of 99,400,891. These efficiency tricks dramatically reduce our search, from about 100 million numbers to only 1,228 (a smaller number than the square root of 99,400,891). For those curious, 99,400,891 = 9,967 × 9,973, so it’s not prime.
Those shortcuts worked wonders for an eight-figure number, but how did Durant get to 41,024,320? To go from pure massive search to truly massive, he and many other search engines focus on specific types of prime numbers. The first Mersenne, Marin Mersenne, XVII. The French theologians studied in the 19th century take a special form. You get them by multiplying 2 by itself and then subtracting 1, as shown in the equation 2.n – 1. Mersenne realized when connecting different values nsometimes you get a prime number. Specifically, 2n – 1 can only give a prime n he is the first, and even then it is not guaranteed. What makes Mersenne’s firsts so special a hunter’s perspective We know a quick method (called Lucas-Lehmer dominance test) to check whether the numbers in form 2n – 1 are the first. This test is much faster than the known general methods for numbers without this special form.
The Lucas-Lehmer test powers the Great Internet Mersenne Prime Search project, which was launched in 1996 and is open to any volunteer. download a free code Mersenne is the first to run on their computers. The crowdsourced approach and focus on Mersenne primes has been successful. The the seven largest known prime numbers all are Mersenne firsts and were discovered by project participants. Note that it is smaller unknown the former certainly exist, but as we know of no effective method of verifying them, they will remain in the shadows for the time being.
All told, project volunteers have discovered 18 new Mersenne firsts, 17 of which owe their discovery to hobbyists’ personal computers. Durant, a 36-year-old former Nvidia engineer, just broke that streak. Nvidia, which recently surpassed Apple as the world’s most valuable companydesigns specialized computer chips called graphics processing units (GPUs). As the name suggests, GPUs were invented to speed up graphics rendering, but they also excel at other tasks involving highly parallel computing, in which many processors run simultaneously to solve problems. These problems include training neural networks such as GPT-4, a cryptocurrency miner, and thus finding primes. Durant assembled a global supercomputer by purchasing processing time from various cloud GPU providers. At its peak, Durant’s project scored 12 times more than all the other computers involved in the first Mersenne search combined.
In addition to heavy hardware, the Mersenne prime search software also received a significant upgrade since the latest discovery. To ensure Mersenne first replaced the super-fast Lucas-Lehmer test with the super-fast. test first test. Given a check number, this last test confirms that the first one is not, or says that it is probably first The first probables have a very small chance of not being the first. Only when a computer finds the first probability do Mersenne search volunteers perform the full Lucas-Lehmer test to remove any doubt. Durant’s first news on October 11 passed the first probable test. Then, on October 19, a year after Durant began searching, independent tests of Mersenne’s first search confirmed that he had found a needle in a haystack: 2136,279,841 – 1 is the largest known prime number.
It surpasses the previous record by more than 16 million figures. If that wasn’t enough glory for him, Durant has also discovered the greatest “perfect number” known. A perfect number is equal to the sum of its divisors (minus itself); 6 is perfect because it is divisible by 1, 2, 3 and is equal to the sum of 1 + 2 + 3. The second perfect number is 28. XVIII The 19th century mathematician Leonhard Euler proved that every perfect even number. can be created From a Mersenne prime, therefore, the discovery of one presupposes a double opposite of mathematical discoveries.
The well could run dry at any moment, however. We don’t know if there is an infinite prime Mersenne number (and therefore also a perfect number). Oddly enough, we don’t know if there are any perfect odd numbers, a question some people call out The oldest unsolved math problem.
When asked how much money his project costs Interview with Numberphile On YouTube, Durant said, “I think it’s under $2 million.” It’s a big investment compared to the $3,000 research project prize, which he plans to give to his high school, the Alabama School of Math and Science. At this point, you may wonder why so many people spend their time and money looking for firsts that have no obvious real-world application. In part, their efforts celebrate human curiosity and serve as benchmarks for our progress in mathematical computation. But I think the founder of the first Mersenne search project, George Woltman, after asking this question In a Numberphile video, he said it best: “It’s fun.”
