It has been nearly six years since the last Mersenne prime was discovered. Now, at last, we have Mersenne prime number 52 and it has 41,024,320 digits!

The new prime is an example of a Mersenne prime, named for Marin Mersenne the French monk who studied them 250 years ago:

Mersenne primes are all of the form 2^p-1, but not all numbers of this form are prime. In particular, if p isn't prime then 2^p-1 isn't prime. For example, 2²-1 or 3 is a prime and so is 2³-1=7, but 2⁴-1 =15 isn't since 4 isn't a prime.

The sequence for p that give primes is 2,3,5,7,13,17,19,31... and now that the series continues with 136279841 being the largest value of p found to date. And, with the latest discovery, we now know 52 values of p for which 2^p-1 is prime.The new entry is named M136279841, as quoting more than 41 million digits is time consuming. 

The Great Internet Mersenne Prime Search, more commonly referred to as GIMPS, has organized the search since 1996 and has found the last 17 Mersenne primes, including this one.

His previous two discoveries, M77232917 and M82589933, came in December 2017 and 2018 respectively with the latter having 24,862,048 digits, more than one and a half million more than its predecessor. The difference on this occasion is more than 16 million digits!

This approach used to find this particular prime is a slight departure from past procedures as it was found using GPUs rather than desktop CPUs. The GPU in this case was an Nvidia H100, presumably given a few days off from training neural networks.

Its discoverer, Luke Durant, an  ex-Nvidia staffer, presumably knows his way around a GPU card. Another difference is that the number was first discovered as prime using a test for probable primeness, see our report Prime Numbers And Primality Testing. This test gave the first indication that a prime had been found on October 11, but given that the test used could be wrong a deterministic algorithm was used on October 12 and confirmed that it was indeed prime.

Let me say again that, as far as I know there are no practical uses for Mersenne Primes and you don't even need to see their digits to do most number theory on them. The only practical spin-off is the proof, if we needed it, that GPUs are amazingly powerful if you have the right sort of problem.

If you really want to see the digits then take a look at the following video:

More Information

GIMPS Discovers Largest Known Prime Number: 2136,279,841-1

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