The new largest prime number is found!
Posted on Oct 22, 2024 / Technology
The management of the non-profit project GIMPS announced the discovery of a new record-breaking prime number.
It turned out to be2¹³⁶²⁷⁹⁸⁴¹ − 1
that is, two to the power of 136279841 minus one.The record number requires 41024320 digits - an archive with them can be downloaded from the link on the project website, it itself takes up about 20 megabytes.
The new largest prime number was found by researcher and former Nvidia employee Luke Durant. To search for it, the enthusiast deployed his own computing cluster in the cloud. Moreover, the calculations were carried out not on processors, but on video cards.
As reported on the GIMPS website, Durant received the first search result, indicating a high probability that the new number is indeed prime, on October 11. The next day, the researcher verified the result using a control algorithm (the Luke-Lehmer test) - it clearly confirmed the primacy of the found number. Then Durant's result was rechecked by several experts using different programs implementing the same algorithm, and the calculations were carried out both on graphics cards and on regular processors. On October 21, GIMPS officially announced the discovery of the new number.
Like the vast majority of record-breaking prime numbers discovered in recent centuries, the new number is a Mersenne prime, meaning it can be represented as a power of two minus one. French mathematician Marin Mersenne made significant contributions to the study of prime numbers in the 17th century, although the numbers themselves had already been discussed by the ancient Greeks.
The earliest Mersenne primes are 1, 3, 7, 15, 31, and 63, or 2¹−1, 2²−1, 2³−1, 2⁴−1, 2⁵−1, 2⁶−1, and so on. As you can see, not all of them are prime — 1, 15, and 63 are not. In addition, not all prime numbers are Mersenne primes — for example, 11, 13, and 17 cannot be represented as powers of two.
Nevertheless, Mersenne primes do prevail in the historical series of record holders — primarily because their search is much easier than searching for prime numbers of other types. There is a faster and more efficient verification algorithm for Mersenne primes — the very same Lucas-Lehmer test, formulated by Edouard Lucas in 1878 and proven by Dick Lehmer in 1930. The complexity of checking the primality of a number using this test increases with the square of the number itself, so searching for new prime numbers requires a multiple increase in computing resources.
Currently, the search for new record-breaking prime numbers is mainly carried out not by university mathematicians, but by GIMPS volunteers. As a result, a dozen and a half records over the past decades belong to this online community. Prime numbers are widely used in some important modern encryption algorithms (for example, RSA), and in this sense they have a specific practical application. However, this does not apply to record-breaking prime numbers with millions of digits in decimal notation; they are not needed for cryptography. Finding them is primarily an aesthetic task, although in the past it has yielded several important results in the theory of algorithms and computer science.
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