Prime numbers play a crucial role in modern cryptography, particularly in the widely-used RSA encryption algorithm12. The security of RSA relies on the difficulty of factoring large numbers that are the product of two prime numbers2. Key aspects of using primes in cryptography include:

• Large primes, often hundreds of digits long, are used to generate public and private keys34.

• The product of two large primes forms the basis of the public key, while the prime factors themselves are kept secret24.

• Generating suitable primes involves using probabilistic primality tests like the Miller-Rabin test56.

• The larger the primes used, the more secure the encryption, as factoring becomes exponentially more difficult with increasing number size34.

The Great Internet Mersenne Prime Search (GIMPS) project has entered a new era with the integration of GPU computing, marking a significant shift from its 28-year reliance on CPUs1. This collaboration has led to the discovery of the largest known Mersenne prime, which is also the first to be found using a GPU21. The breakthrough was achieved using NVIDIA's A100 GPU and confirmed on the more powerful H100 GPU, demonstrating the immense potential of graphics processors in mathematical computations1.

• GIMPS now offers a suite of programs for various CPUs and GPUs, including Mihai Preda's groundbreaking GpuOwl program3.

• The project allows volunteers to contribute computing power, similar to distributed computing initiatives like Folding@Home1.

• GPU-based searches have proven to be significantly faster and more efficient for prime number calculations23.

• This advancement not only pushes the boundaries of mathematics but also serves as a stress test for GPUs, showcasing their evolving capabilities in scientific computing1.

Mersenne primes are a special class of prime numbers that take the form 2^p - 1, where p is also a prime number1. Named after the 17th-century French mathematician Marin Mersenne, these numbers have fascinated mathematicians for centuries due to their unique properties and rarity2. Some key characteristics of Mersenne primes include:

• They are closely connected to perfect numbers, as every Mersenne prime generates an even perfect number3.

• The search for Mersenne primes has become a major focus of distributed computing projects like GIMPS (Great Internet Mersenne Prime Search)24.

• As of 2024, only 52 Mersenne primes are known, with the largest having 41,024,320 decimal digits4.

• Mersenne primes are extremely rare and become increasingly sparse as the exponent p increases1.

• While not having many practical applications, the search for Mersenne primes has driven advancements in computer hardware and software4.

The discovery of M136279841, the largest known Mersenne prime as of October 2024, marks a significant milestone in the field of mathematics and computational research. This prime number, with 41,024,320 digits, was found on October 12, 2024, by Luke Durant, a researcher and former Nvidia employee from San Jose, California12.

Key aspects of this discovery include:

• It's the first Mersenne prime with an exponent surpassing 8 digits (136,279,841)3.

• The prime was initially identified as probably prime by an Nvidia A100 GPU in Dublin and then confirmed by an Nvidia H100 in San Antonio, Texas2.

• This discovery represents the first time a Mersenne prime was found using a probable prime test, sparking debate about the official discovery date1.

• The new prime generates a perfect number with more than 82 million digits2.

• This discovery ended the previous record holder's (M82589933) reign of over 6 years, which was the longest since M19937 held the record from 1971 to 19784.

The discovery of M136279841 not only pushes the boundaries of known prime numbers but also demonstrates the evolving capabilities of GPU technology in mathematical computations2.