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GIMPS Project Discovers

Largest Known Prime Number: 2^{77,232,917}-1

**RALEIGH, NC., January 3, 2018** -- The
Great Internet Mersenne Prime Search (GIMPS) has discovered the
largest known prime number, 2^{77,232,917}-1, having
23,249,425
digits. A computer volunteered by Jonathan Pace made the find on
December 26, 2017. Jonathan is one of thousands of volunteers using
free GIMPS software available at www.mersenne.org/download/.

The new prime number, also known as M77232917, is calculated by multiplying together 77,232,917 twos, and then subtracting one. It is nearly one 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 50th known Mersenne prime ever discovered, each increasingly 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 16 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.

The primality proof took six days of non-stop computing on a PC with an Intel i5-6600 CPU. To prove there were no errors in the prime discovery process, the new prime was independently verified using four different programs on four different hardware configurations.

- Aaron Blosser verified it using Prime95 on an Intel Xeon server in 37 hours.
- David Stanfill verified it using gpuOwL on an AMD RX Vega 64 GPU in 34 hours.
- Andreas Höglund verified the prime using CUDALucas running on NVidia Titan Black GPU in 73 hours.
- Ernst Mayer also verified it using his own program Mlucas on 32-core Xeon server in 82 hours. Andreas Höglund also confirmed using Mlucas running on an Amazon AWS instance in 65 hours.

Jonathan Pace is a 51-year old Electrical Engineer living in Germantown, Tennessee. Perseverance has finally paid off for Jon - he has been hunting for big primes with GIMPS for over 14 years. The discovery is eligible for a $3,000 GIMPS research discovery award.

GIMPS Prime95 client software was developed by founder George Woltman. Scott Kurowski wrote the PrimeNet system software that coordinates GIMPS' computers. Aaron Blosser is now the system administrator, upgrading and maintaining PrimeNet 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 Jonathan Pace 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 "J. Pace, G. Woltman, S. Kurowski, A. Blosser, et al."

**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 2^{P}-1.
The first Mersenne primes are 3, 7, 31, and 127 corresponding to
P = 2, 3, 5, and 7 respectively. There are now 50 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 Mersenne prime discoveries were made by members
in various countries.

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 2^{77,232,916} x (2^{77,232,917}-1). This
number is over 46 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. David Stanfill's and Ernst Mayer's verification computations for this discovery was donated by Squirrels LLC (http://www.airsquirrels.com) which services K-12 education and other customers.

^{[1]}

*Science (American Association for the Advancement of Science), May 6, 2005 p810.*