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https://www.mersenne.org/primes/?press=M136279841
The Great Internet Mersenne Prime Search (GIMPS) has discovered the largest known prime number, 2136,279,841-1, having 41,024,320 decimal digits. Luke Durant, from San Jose, California, found the prime on October 12th.
The new prime number, also known as M136279841, is calculated by multiplying together 136,279,841 twos, and then subtracting 1. It is over 16 million digits larger than the previous record prime number, in a special class of extremely rare prime numbers known as Mersenne primes. It is only the 52nd known Mersenne prime ever discovered, each increasingly more difficult to find. Mersenne primes were named for the French monk Marin Mersenne, who studied these numbers more than 350 years ago. GIMPS, founded in 1996, has discovered the last 18 Mersenne primes. Volunteers download a free program to search for these primes, with a $3000 award offered to anyone lucky enough to find a new prime. Prof. Chris Caldwell founded an authoritative web site on the largest known primes which is now maintained by volunteers, and has an excellent history of Mersenne primes.
(Score: 3, Troll) by ledow on Thursday October 24 2024, @09:09AM (2 children)
If you're sitting on an secured TLS/SSL website and asking how discovering new large prime numbers can be "useful", you really need to go read even the most basic textbook on cryptography.
(Score: 3, Touché) by EJ on Thursday October 24 2024, @03:09PM
I neither need nor want to be an expert in everything. Simply saying "cryptography" was good enough to satisfy my question.
(Score: 1) by pTamok on Friday October 25 2024, @12:44PM
Not really: Mersenne primes are not used in the encryption methods in TLS/SSL unless it is by accident. They are far too well known, there are not many of them, and it would be easy to check a list of them, because it would be short. They are also too big for RSA - e.g. 1024-bit RSA requires two primes that are roughly 512 bits long (the two primes should be roughly the same size); RSA-2048 two primes of roughly 1024 bits, and RSA-4096 two primes of roughly 2048 bits. The newly found Mersenne prime has 100s of thousands of bits. Too big.
However, RSA does use primes. The prime number theorem states that the probability that an integer chosen at random, k, is prime is approximately 1/ln(k). Theis means that for numbers in the range of 2048 bits, about 1 in 1400 numbers are likely to be prime*. So you get a good random number generator, and get it to generate about that many numbers of 2048 bits, and start testing them to see if they are prime. For this, you use the Miller-Rabin primality test algortirm, which gives you a probabilistic answer as it would take far too long to factor the number to see if it were prime.
More details in this blog: How to find large prime numbers for RSA with the Miller-Rabin Primality Test [incolumitas.com]. They main bit is in the section headed: "Generating large prime numbers p and q for RSA"
RSA encryption uses prime numbers of about 512, 1024, and 2048 bits. It doesn't need numbers larger than about 2048 bits, so generating new primes with 100s of thousands of bits doesn't help.
* a 2048-bit number is of the order of 22048. The natural logarithm of 22048 is 2048 x ln(2), which is 2048 x 0.693 = 1420 or 'about' 1400.