Thousands of computers around the world are currently scouring the number line in a treasure hunt for rare mathematical gems. Prime number enthusiasts, seeking larger and larger numbers that are only divisible by 1 and themselves, marshal vast computing power and algorithmic ingenuity in hopes of engraving their names in manuscripts of the history of mathematics.

The final entry comes from Luke Durant, a researcher from San Jose, California. Durant’s discovery toppled the previous record holder, who remained unchallenged for nearly six years, a reign of unprecedented length in modern research. ever-increasing prime numbers. This gap makes sense: as prime numbers increase, they move further apart, making each new discovery more difficult than the last.

The new prime number contains a mind-boggling 41,024,320 digits. To put this into perspective, the estimated number of atoms in the observable universe it displays around 80 digits. Each additional digit increases a number by 10 times, so that the size of the new prime number far exceeds human intelligibility. Prime numbers play a major role in pure mathematicswhere these are the main characters of a field called number theoryand in practice, where, for example, they underlie encryption algorithms. A prime number with 41 million digits will not immediately join the ranks of useful figures, but for now, it adds a feather to the cap of a community that aspires to grasp the colossal.

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Durant’s success comes in part from the team’s clever new software. Excellent Internet search Mersenne Prime and in part due to the robust hardware and computing power he personally mobilized for this pursuit. By assembling a “cloud supercomputer” spanning 17 countries, he ended a long tradition of discovering personal computers through prime numbers.

Prime numbers are often called the “building blocks of mathematics” because every integer greater than 1 has a fingerprint that is the product of a unique collection of prime numbers. For example, 15 is the product of the prime numbers 5 and 3, while 13 cannot be subdivided like this because 13 is prime. The study of these numbers dates back at least to the ancient Greeks. In 300 BCE Euclid proven in his manual Elements that there are infinitely many prime numbersand mathematicians, professionals and amateurs alike, have rejoiced in their hunt ever since.

Although the first string of prime numbers (2, 3, 5, 7, 11, etc.) is easy to find, the task becomes considerably more difficult as the numbers increase. For millennia, people found prime numbers by hand, until 1951, when computers took over the search. But even silicon bounty hunters have trouble spotting prime numbers at the ends of the number line, because testing the primality of a huge number is non-trivial. To deal with this, researchers deploy every little optimization trick possible to speed up their tests or narrow their hunting ground, thus increasing their chances of finding a new prime number.

Consider the number 99,400,891. How would you determine whether it is prime or not? You can simply divide it by each smaller number and check if it has any divisors (besides 1 and itself). But that’s almost 100 million cases to check a relatively small eight-digit number. You would save a lot of work by realizing that you don’t need to check every number up to the target, just the prime Numbers. For what? You just need to find a single divisor (a number that properly divides 99,400,891 with no remainder). We know that any non-prime divisor can be decomposed into prime factors: if your target is divisible by 15, then it is also divisible by the prime numbers 5 and 3, so you just need to check the latter to determine primality. Additional savings would come from the fact that you also don’t need to check all the smaller prime numbers, only those up to the square root of 99,400,891 (the number which, when multiplied by itself, gives you this eight-figure result). If none of these smaller prime numbers divide it cleanly, then you can stop looking, because the product of two numbers greater than the square root of 99,400,891 will exceed it. These efficiency tricks significantly reduce our search, from about 100 million numbers to just 1,228 (the number of primes less than the square root of 99,400,891). For the curious, 99,400,891 = 9,967 × 9,973, so it’s not prime.

These shortcuts worked wonders for an eight-digit number, but how did Durant reach 41,024,320 digits? To take the research from simply massive to truly gargantuan, he and many other researchers focus on particular types of prime numbers. The Mersenne bounties, named after Marin Mersenne, the French theologian who studied them in the 17th century, take a particular form. You get them by multiplying 2 by itself a certain number of times and then subtracting 1, as described in equation 2.ⁿ – 1. Mersenne noticed that when you plug in different values ​​for nwe sometimes obtain a prime number. More precisely, 2ⁿ – 1 can only give a prime number when n itself is prime, and even then it’s not guaranteed. What makes Mersenne premiums special the point of view of a great hunter is that we know a quick method (called Lucas-Lehmer primality test) to check if numbers of the form 2ⁿ – 1 is prime. This test is much faster than any known general method for numbers without this special form.

The Lucas-Lehmer test powers the Great Internet Mersenne Prime Search project, launched in 1996 and allows any volunteer to download a free code who are looking for Mersenne bounties to run on their computers. Mersenne’s participatory approach and emphasis on incentives have proven effective. THE seven largest known prime numbers are all Mersenne primes and were all found by project participants. Note that smaller unknown Prime numbers certainly exist, but since we don’t know effective methods to verify them, they will remain in the shadows for the moment.

In total, project volunteers found 18 new Mersenne prime numbers, 17 of which owe their discovery to amateur personal computers. Durant, a 36-year-old former Nvidia engineer, has just put an end to this streak. Nvidia, which recently briefly overtook Apple as global leader most valuable companydesigns specialized computer chips called graphics processing units (GPUs). As their name suggests, GPUs were originally invented to speed up the rendering of graphics, but they also excel at other tasks involving highly parallelized calculations, in which many processors work simultaneously to solve problems. These problems include training neural networks like GPT-4, mining cryptocurrency, and ultimately finding prime numbers. Durant assembled a global supercomputer by purchasing processing time from various cloud GPU providers. At its peak, Durant’s project generated about 12 times more numbers than all the other computers involved in Mersenne’s primary research combined.

In addition to the robust hardware, the Mersenne Prime research software has also received a notable upgrade since the last discovery. He replaced the ultra-fast Lucas-Lehmer test to certify Mersenne prime numbers with an ultra-duper-fast test probable main test. Given a number to check, this last test either confirms that it is not prime, or indicates that it is prime. probably prime. Probable prime numbers have a very small chance of turning out to be non-prime. Only once a computer finds a likely prime number do Mersenne’s volunteers perform the full Lucas-Lehmer test to remove any doubt. Durant’s new prime passed the probable prime test on October 11. Then, on October 19, a year after Durant’s search began, independent tests by Mersenne’s bounty search confirmed that he had indeed found a needle in a haystack:2^136,279,841 – 1 is the largest known prime number.

It surpasses the previous record holder by more than 16 million digits. If that didn’t bring him enough fame, Durant also discovered the largest known “perfect number.” A perfect number is equal to the sum of its divisors (excluding 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. The 18th century mathematician Leonhard Euler proved that every even perfect number can be generated of a Mersenne prime, so finding one promises a two-for-one agreement on mathematical discoveries.

But the well could dry up at any time. We do not know if there exists an infinite number of Mersenne primes (and therefore even perfect numbers). Interestingly, we don’t know if there are odd perfect numbers, a question some call the oldest unsolved math problem.

When asked how much money his project cost in a interview with Numberphile on YouTube, Durant said: “I believe it’s less than $2 million. » It’s a considerable investment compared to the Prime Search project’s $3,000 price tag, which he plans to donate to the high school he attended, the Alabama School of Mathematics and Science. At this point, you may be wondering why so many people spend their time and money pursuing core values ​​that have no obvious real-world applications. Their efforts celebrate human curiosity in part and serve as a benchmark for our progress in mathematical computation. But I think that the founder of the Mersenne Prime Search project, George Woltman, when asked this question in a Numberphile video, he said it best: “It’s fun.”