A New Mersenne Prime Discovery
Written by Sue Gee
Saturday, 29 December 2018

The 51st Mersenne Prime was discovered on December 7, 2018, less than a year after the previous one was found on December 26, 2017. The new prime number, known as M82589933, has a value of 2^82,589,933-1. It has 24,862,048 digits, more than one and a half million more than its predecessor.

When he reported on the discovery of M77232917, see Largest Prime Now Has Over 23 Million Digits, almost a year ago Mike James provided a good explanation of both prime numbers and Mersenne primes and why they are important. He also noted with regard to large prime numbers:

There are no practical uses for Mersenne primes or primes of such size. They are too big for cryptography and most of the work in number theory about them doesn't require actual examples of numeric values.

Not having any practical use doesn't stop people hunting for Mersenne primes and, like the previous one, the new one comes from GIMPS, the Great Internet Mersenne Prime Search, which provides free software to thousands of volunteers who are engaged in the search.

In its press release for M82589933 GIMPS gave details of the volunteer responsible for the latest find, who wins \$3,000 for its discovery:

Patrick Laroche is a 35 year old I.T. professional living in Ocala, Florida. For many years, Patrick had used GIMPS software as a free "stress test" for his computer builds. Recently, he started prime hunting on his media server to "give back" to the project. After less than 4 months and on just his fourth try, he discovered the new prime number. By way of comparison, some GIMPS participants have searched for more than 20 years with tens of thousands of attempts but no success. Thus proving that even the "little guy" can compete against those with lots of computing resources.

The press release also states:

GIMPS has also been extremely lucky over the last 15 years. This is GIMPS' 12th prime discovery between 220000000-1 and 285000000-1, triple the expected number of new primes. One reason to search for new primes is to match actual results with expected results. This anomaly is not necessarily evidence that existing theories on the distribution of Mersenne primes is incorrect. However, if the trend continues it may be worth further investigation.

Once a Mersenne prime (named for Marin Mersenne the French monk who studied them 250 years ago) has been identified it has to be proved and verified. Although verification is a much quicker process it is still time- (compute time-) consuming.

According to the GIMPS press release, the primality proof took twelve days of non-stop computing on a machine with an Intel i5-4590T CPU. To prove there were no errors in the prime discovery process, the new prime was independently verified using three different programs on three different hardware configurations.

• Andreas Höglund verified it using CUDALucas running on a NVidia V100 GPU in 21 hours.
• Andreas Höglund also verified it using Mlucas running on 16 cores of an Amazon AWS instance in 72 hours.
• Aaron Blosser also verified it using Prime95 on an Intel 7700K processor in 6 days, 8 hours.

GIMPS' next major goal is to win the \$150,000 award administered by the Electronic Frontier Foundation offered for finding a 100 million digit prime number.

Will this happen within the next 12 months?

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