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Around October of 2024, you may have heard that a new prime had been discovered. I myself heard about it from all sorts of news sources. The project that found the new prime is called GIMPS (Great Internet Mersenne Prime Search), and it’s actually a program downloadable and usable by anyone. It uses the idle computing time of computers who choose to download it to verify larger and larger instances of a particular type of prime, a Mersenne prime.
Mersenne primes occur in the form 2p - 1, where p is a prime. You can see this in action by plugging in small primes. Beginning from p=2, it generates 3, 7, 31, 127. The form seems almost magical in its ability to generate primes, but when p=11, it generates 2047, which equals 23*89. After this original string of successes, the number of Mersenne primes starts to increase far slower. As of 2025, only 52 Mersenne primes are known.
Mersenne primes are special in a notable way: they get massive quickly, and they’re easier to check for primality than other potential primes. Testing for primality often revolves around using the factorization of n-1 or n+1, but for Mersenne primes, n+1 is easy: a power of two. The specific method used for Mersenne primes is called the Lucas-Lehmer test, and it’s significantly more efficient than the general test. For this reason, the 7 largest primes we know are Mersenne primes.
Now, the thing I wanted to talk about today is the “practical” use of this new prime. In many of the articles I read about the prime, I hear them referencing RSA encryption, how it uses primes to keep your data safe, and how this new prime will increase that security. Now, this is simply not the case. I’ve discussed the mechanisms of RSA encryption in a previous article, and while it is true that it relies on the difficulty of factoring the product of two large primes, this new 41-million digit prime will certainly not be used in encryption.
The thing is, cracking RSA with primes of only 300 digits is already a problem far too difficult for even our best computers. Until quantum computers roll around to crack open these problems, our current system is more than secure. When that time does come when quantum computers can break the internet, it may be time to change the system entirely rather than just getting bigger numbers. To use this ginormous prime in a system where something thousands of magnitudes smaller than it suffices is… overkill. Actually, it’s more than overkill. It would be counterproductive to your purpose of security. One major factor that adds to the difficulty of cracking RSA is the fact that there are trillions upon trillions of 300 digit primes to choose. To contrast, there are only 2964 known primes with more than 1 million digits. As such, cracking RSA that used giant primes, like our new friend, would be massively easier than cracking traditional RSA.
You might wonder then, why are we discovering these giant primes? Simply put, the answer is curiosity. You could probably call it fun. We’re doing it for the sake of doing it, and for the sake of being able to say we could. If that does sound fun, download GIMP! And seriously, do not use this new prime for encryption!
https://www.mersenne.org/primes/?press=M136279841
https://en.wikipedia.org/wiki/Largest_known_prime_number