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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: 2) by EJ on Thursday October 24, @02:01AM (10 children)
How is this useful? Can you even do anything meaningful with such a large number? Aside from simple curiosity, is there any value in knowing this, like knowing the quadrillionth digit of pi.
(Score: 1, Funny) by Anonymous Coward on Thursday October 24, @02:56AM (2 children)
> How is this useful?
"Utility" is a funny concept with widely varying definitions among people. I'm guessing that "math bragging rights" might have something to do with it. But then there is the cash too--from tfs,
> Volunteers download a free program to search for these primes, with a $3000 award offered to anyone lucky enough to find a new prime.
(Score: 2) by EJ on Thursday October 24, @03:32AM
Well, yeah. I'm just thinking in terms of something like, "if only we could find a large enough prime, encryption would be unbreakable or we might understand how black holes work."
(Score: 4, Touché) by davidjohnpaul on Thursday October 24, @05:00AM
The guy who discovered it is "pretty sure" he spent less than $2 million on computing resources to find this one, so I don't think the $3000 was much of a motivator in this case.
(Score: 1, Informative) by Anonymous Coward on Thursday October 24, @04:10AM (1 child)
Large primes are used for data encryption and the like.
(Score: 3, Funny) by DannyB on Thursday October 24, @07:42PM
"That is true!" explained the senator. "We have now devised a ROT17 encryption which requires a different operation to decrypt than ROT13."
"Furthermore", explained the senator, "we chose 17 because it is a prime number, unlike 13."
Santa/Satan maintains a database and does double verification of it.
(Score: 3, Insightful) by pTamok on Thursday October 24, @07:41AM (1 child)
It allows you to determine another perfect number.
https://math.libretexts.org/Bookshelves/Combinatorics_and_Discrete_Mathematics/Elementary_Number_Theory_(Barrus_and_Clark)/01%3A_Chapters/1.16%3A_Perfect_Numbers_and_Mersenne_Primes [libretexts.org]
There is also a proposed 'post-Quantum' cryptosystem based upon the use of Mersenne primes: Mersenne-756839 (PDF) [nist.gov]
It has already been 'successfully' attacked. Exploiting Decryption Failures in Mersenne Number Cryptosystems [kuleuven.be], success being defined as:
But the thing to remember is that attacks tend to get better over time.
(Score: 2) by DannyB on Thursday October 24, @07:47PM
There are perfect numbers. But there are no perfect prime ministers.
A perfect prime minister would be one whose length of tenure was equal to the sum of its proper divisors. But since the minister is prime it doesn't have any proper divisors other than one. Thus there are no perfect prime ministers.
Santa/Satan maintains a database and does double verification of it.
(Score: 3, Troll) by ledow on Thursday October 24, @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, @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, @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.