Over 41 million decimal places
24 October 2024
By Elke von Rekowski
Reading time: about 3 minutes
Using a global GPU system, researcher Luke Durant has found the largest prime number known to date: 2136,279,841–1. This number is over 41 million decimal places long and belongs to the class of Mersenne prime numbers, which have been studied by mathematicians for centuries. But why is it so difficult to keep discovering new prime numbers, and why is this especially true for the Mersenne prime numbers?
Prime numbers play a fundamental role in number theory.
Foto: picture alliance / Science Photo Library
Prime numbers are numbers that can only be divided by themselves and by 1. This property has intrigued mathematicians even in ancient times, because prime numbers play a fundamental role in number theory. However, while the first prime numbers such as 2, 3, 5 or 7 are easy to recognize, it becomes increasingly difficult to recognize larger prime numbers. That’s because Prime numbers The natural numbers are irregularly distributed and there is no simple formula to predict them.
Mersenne primes are a special subgroup of primes of the form 2p–1, where the exponent pa must itself be a prime number. What is special about these numbers is their mathematical simplicity combined with extraordinary magnitude. The French monk Marin Mersenne studied the first prime Mersenne numbers over 350 years ago, which bears his name today. Despite their seemingly simple structure, Mersenne primes are extremely rare – only 52 have been discovered so far.
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We have a new record!
The Great Internet Mersenne Prime Search verified that this supernet is indeed excellent.
Which also means that a new perfect number has been found.
The last record was held for an unusually long time – almost 6 years. We’ll see how long this lasts. pic.twitter.com/LB4zLGbZuq
— Jay Cummings (@LongFormMath) October 21, 2024
The difficulty of finding prime numbers
Finding new Mersenne prime numbers is a huge computational challenge. As the exponent p increases, the candidates grow exponentially. Calculating even 2136,279,841 is a huge mathematical achievement that requires special algorithms and modern hardware. The The GIMP project (Great Internet Mersenne Prime Search), dedicated to this research since 1996, uses distributed computing to generate the massive computing power needed to find larger prime numbers.
The main difficulty in finding prime numbers is the necessary proof. It is not enough to run an algorithm that identifies potential prime numbers – each number found must also be confirmed with special mathematical tests such as the Lucas-Lehmer test.
Furthermore, the non-uniformity of the distribution makes it difficult to search for prime numbers. Although the distances between the first prime numbers are relatively small, they increase as the size of the numbers increases. This means that it becomes more likely to find a new prime number in a given range of natural numbers. This is especially true for Mersenne prime numbers, where the intervals between discoveries are often several years.
A complex and computationally intensive proof
In contrast to smaller prime numbers, which can be checked using relatively simple procedures, the huge numbers that arise when searching for Mersenne primes require special mathematical methods. The Lucas-Lehmer test, developed by Édouard Lucas and Derrick Henry Lehmer in 1930, is the central test to confirm Mersenne’s prime. It only works for numbers of the form 2p–1, which makes it ideal for checking these special prime numbers.
The test is based on an iterative sequence calculated specifically for the exponent p of the Mersenne prime. If the last term of this sequence is zero, it is actually a prime number. However, although this test is more efficient than other prime number detection methods, very large numbers like 2136,279,841–1 require enormous computing resources. Each calculation can take weeks or months, and to ensure that no errors occur, multiple independent tests must be performed.
Future prime number search
With the discovery of 2136,279,841–1, GPUs (graphics processing units) took on a leading role in prime number searches. Previously, CPUs (Central Processing Units) made such discoveries, but the huge increase in the computing power of GPUs, especially in cloud infrastructure, now allows much larger numbers to be tested. Luke Durant, the finder of the current prime number, used thousands of GPUs spread across many countries (distributed computing) to achieve this feat.
But what is the use of such a discovery? So far, those huge prime numbers are of purely theoretical interest. However, that may change in the future, as the search for them could drive the development of applications in cryptography and other areas of information technology.