Mersenne Prime Number discovery - 2^82589933-1 is Prime!

The Great Internet Mersenne Prime Search (GIMPS) has discovered the largest known prime number, 282,589,933-1, having 24,862,048 digits. A computer volunteered by Patrick Laroche made the find on December 7, 2018. Patrick is one of thousands of volunteers using free GIMPS software available at www.mersenne.org/download/.

The new prime number, also known as M82589933, is calculated by multiplying together 82,589,933 twos, and then subtracting one. It is more than one and a half 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 51st 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 17 Mersenne primes. Volunteers download a free program to search for these primes, with a cash award offered to anyone lucky enough to find a new prime. Prof. Chris Caldwell maintains an authoritative web site on the largest known primes, and has an excellent history of Mersenne primes.

A Lucky Find in a Lucky Streak

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.

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.

Details on the Discovery and Verification

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 the prime using CUDALucas running on a NVidia V100 GPU in 21 hours.
Andreas Höglund also verified the prime 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 Prime95 client software was developed by founder George Woltman. Scott Kurowski wrote the PrimeNet system software that coordinates GIMPS’ computers. Aaron Blosser is the system administrator, upgrading and maintaining the PrimeNet server as needed. Volunteers have a chance to earn research discovery awards of $3,000 or $50,000 if their computer discovers a new Mersenne prime. 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.

Credit for this prime goes not only to Patrick Laroche for running the Prime95 software, Woltman for writing the software, Kurowski and Blosser for their work on the Primenet server, but also the thousands of GIMPS volunteers that sifted through millions of non-prime candidates. In recognition of all the above people, official credit for this discovery goes to “P. Laroche, G. Woltman, S. Kurowski, A. Blosser, et al.”

Patrick’s find is eligible for the $3,000 GIMPS research discovery award.

About Mersenne.org’s Great Internet Mersenne Prime Search

The Great Internet Mersenne Prime Search (GIMPS) was formed in January 1996 by George Woltman to discover new world record size Mersenne primes. In 1997 Scott Kurowski enabled GIMPS to automatically harness the power of thousands of ordinary computers to search for these “needles in a haystack”. Most GIMPS members join the search for the thrill of possibly discovering a record-setting, rare, and historic new Mersenne prime. The search for more Mersenne primes is already under way. There may be smaller, as yet undiscovered Mersenne primes, and there almost certainly are larger Mersenne primes waiting to be found. Anyone with a reasonably powerful PC can join GIMPS and become a big prime hunter, and possibly earn a cash research discovery award. All the necessary software can be downloaded for free at www.mersenne.org/download/. GIMPS is organized as Mersenne Research, Inc., a 501(c)(3) science research charity. Additional information may be found at www.mersenneforum.org and www.mersenne.org; donations are welcome.

For More Information on Mersenne Primes

Prime numbers have long fascinated both amateur and professional mathematicians. An integer greater than one is called a prime number if its only divisors are one and itself. The first prime numbers are 2, 3, 5, 7, 11, etc. For example, the number 10 is not prime because it is divisible by 2 and 5. A Mersenne prime is a prime number of the form 2P-1. The first Mersenne primes are 3, 7, 31, and 127 corresponding to P = 2, 3, 5, and 7 respectively. There are now 51 known Mersenne primes.

Mersenne primes have been central to number theory since they were first discussed by Euclid about 350 BC. The man whose name they now bear, the French monk Marin Mersenne (1588-1648), made a famous conjecture on which values of P would yield a prime. It took 300 years and several important discoveries in mathematics to settle his conjecture.

At present there are few practical uses for this new large prime, prompting some to ask “why search for these large primes”? Those same doubts existed a few decades ago until important cryptography algorithms were developed based on prime numbers. For seven more good reasons to search for large prime numbers, see here.

Previous GIMPS discoveries

Previous GIMPS Mersenne prime discoveries were made by members in various countries.
In December 2017, Jonathan Pace et al. discovered the 50th known Mersenne prime in the U.S.
In January 2016, Curtis Cooper et al. discovered the 49th known Mersenne prime in the U.S.
In January 2013, Curtis Cooper et al. discovered the 48th known Mersenne prime in the U.S.
In April 2009, Odd Magnar Strindmo et al. discovered the 47th known Mersenne prime in Norway.
In September 2008, Hans-Michael Elvenich et al. discovered the 46th known Mersenne prime in Germany.
In August 2008, Edson Smith et al. discovered the 45th known Mersenne prime in the U.S.
In September 2006, Curtis Cooper and Steven Boone et al. discovered the 44th known Mersenne prime in the U.S.
In December 2005, Curtis Cooper and Steven Boone et al. discovered the 43rd known Mersenne prime in the U.S.
In February 2005, Dr. Martin Nowak et al. discovered the 42nd known Mersenne prime in Germany.
In May 2004, Josh Findley et al. discovered the 41st known Mersenne prime in the U.S.
In November 2003, Michael Shafer et al. discovered the 40th known Mersenne prime in the U.S.
In November 2001, Michael Cameron et al. discovered the 39th Mersenne prime in Canada.
In June 1999, Nayan Hajratwala et al. discovered the 38th Mersenne prime in the U.S.
In January 1998, Roland Clarkson et al. discovered the 37th Mersenne prime in the U.S.
In August 1997, Gordon Spence et al. discovered the 36th Mersenne prime in the U.K.
In November 1996, Joel Armengaud et al. discovered the 35th Mersenne prime in France.

Euclid proved that every Mersenne prime generates a perfect number. A perfect number is one whose proper divisors add up to the number itself. The smallest perfect number is 6 = 1 + 2 + 3 and the second perfect number is 28 = 1 + 2 + 4 + 7 + 14. Euler (1707-1783) proved that all even perfect numbers come from Mersenne primes. The newly discovered perfect number is 282,589,932 x (282,589,933-1). This number is over 49 million digits long! It is still unknown if any odd perfect numbers exist.

There is a unique history to the arithmetic algorithms underlying the GIMPS project. The programs that found the recent big Mersenne primes are based on a special algorithm. In the early 1990’s, the late Richard Crandall, Apple Distinguished Scientist, discovered ways to double the speed of what are called convolutions — essentially big multiplication operations. The method is applicable not only to prime searching but other aspects of computation. During that work he also patented the Fast Elliptic Encryption system, now owned by Apple Computer, which uses Mersenne primes to quickly encrypt and decrypt messages. George Woltman implemented Crandall’s algorithm in assembly language, thereby producing a prime-search program of unprecedented efficiency, and that work led to the successful GIMPS project.

School teachers from elementary through high-school grades have used GIMPS to get their students excited about mathematics. Students who run the free software are contributing to mathematical research.”

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