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## Large Prime Number Found by SGI/Cray Supercomputer

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text by: Landon Curt Noll

## Note: This is no longer the largest known prime

The largest known prime number may be found in chongo's table of
Mersenne Prime Digits and Names.   <<== try me

EAGAN, Minn., September 3, 1996 -- Computer scientists at SGI's former Cray Research unit, have discovered a large prime number while conducting tests on a CRAY T90 series supercomputer.

The prime number has 378,632 digits. Printed in newspaper-sized type, the number would fill approximately 12 newspaper pages.

In mathematical notation, the new prime number is expressed as 2^1257787-1 , which denotes two, multiplied by itself 1,257,787 times, minus one. Numbers expressed in this form are called Mersenne prime numbers after Marin Mersenne, a 17th century French monk who spent years searching for prime numbers of this type. See Chris Callwell's prime page for more information on prime numbers.

Prime numbers can be divided evenly only by themselves and one. Examples include 2, 3, 5, 7, 11 and so on. The Greek mathematician Euclid proved that there are an infinite number of prime numbers. But these numbers do not occur in a regular sequence and there is no formula for generating them. Therefore, the discovery of new primes requires randomly generating and testing millions of numbers.

Silicon Graphics employees have been at the leading edge for prime number discoveries since 1978. The 10 of the last 11 records for the largest known prime belong to people who are now working at Silicon Graphics:
```Prime Digits Year SGI Employee

2^21701-1 6533 1978 Landon Curt Noll
(with Laura Nickel, now Ariel Glenn)
2^23209-1 6987 1979 Landon Curt Noll
2^44497-1 13395 1979 David Slowinski
(with Harry Nelson)
2^86243-1 25962 1982 David Slowinski
2^132049-1 39751 1983 David Slowinski
2^216091-1 65050 1985 David Slowinski
391581 * 65087 1989 Landon Curt Noll
2^216193-1 John Brown
Sergio Zarantonello
(with Joel & Gene Smith, Bodo Parady)
2^756839-1 227832 1992 David Slowinski
Paul Gage
2^859433-1 258716 1994 David Slowinski
Paul Gage
2^1257787-1 378632 1996 David Slowinski
Paul Gage
```

"Finding these special numbers is a true 'needle-in-a-haystack' exercise, but we improve our odds by using a tremendously fast computer and a clever program," said David Slowinski, a Cray Research computer scientist. and fellow Cray Research computer scientist Paul Gage developed the program that found the this large prime number. Mathematician Richard Crandall independently verified that the number Slowinski and Gage found is prime.

Prime numbers have applications in cryptography and computer systems security. Huge prime numbers like those discovered most recently are principally mathematical curiosities, but the process of searching for prime numbers does have several practical benefits.

For instance, the "prime finder" program developed by Slowinski and Gage is used by SGI's former Cray Research unit, as a quality assurance test on all new supercomputer systems. A core element of this program is a routine that involves squaring a number repeatedly. As this process continues, it eventually involves multiplying immense numbers -- numbers of hundreds of thousands of digits -- by themselves.

"This acts as a real 'torture test' for a computer," said Slowinski. "The prime finder program rigorously tests all elements of a system -- from the logic of the processors, to the memory, the compiler and the operating and multitasking systems. For high performance systems with multiple processors, this is an excellent test of the system's ability to keep track of where all the data is." Slowinski said the recent CRAY T90 series supercomputer test in which this prime number was discovered would run for over 6 hours on one central processing unit of the system. "If a machine can complete this exhaustive run-through, we can be confident everything is working as it should," said Slowinski.

In addition, Slowinski said, techniques used to speed up the performance of the prime finder can also be used to enhance the performance of programs customers use on real-world problems such as forecasting the weather and searching for oil. "Through our work on the prime finder program, we learn new techniques for speeding up certain kinds of mathematical operations. These operations are often key elements of the most computation-intensive portions of software programs our customers run on their systems," said Slowinski.

Slowinski compared running the prime finder on supercomputers and continually "tuning" the program to building and racing exotic cars. "There aren't many practical uses for dragsters or Formula 1 race cars. But some things engineers do to make those cars perform better eventually find their way into cars you and I drive," said Slowinski.

Slowinski noted that with the discovery of this prime number, a new perfect number can also be generated. A perfect number is equal to the sum of its factors. For example, 6 is perfect because its factors -- 1, 2 and 3 -- when added together, equal 6. Mathematicians don't know how many perfect numbers exist. They do know, however, that all perfect numbers have a direct relationship to Mersenne primes. The new perfect number generated with the new Mersenne prime is the 34th known perfect number and has 757,263 digits. The 33rd known perfect number and has 517,430 digits.

Silicon Graphics is participating in the Great Internet Mersenne Prime Search, a group of people all over the world are using George Woltman's program in an orchestrated search for the next Mersenne Prime. Slowinski has verified many of the interim results that the Internet group has sent him.

SGI's former Cray Research unit, creates the most powerful, highest quality computational tools for solving the world's most challenging scientific and industrial problems.

Awarded the cool site of the hour for 9 Sep 1996, 23:00 by cool central.

We were featured on picks from Yahoo!

This Web page was authored by: Landon Curt Noll (chongo <was here> /\oo/\)
on SGI hardware.

``Due to the nature of Mersenne Primes, I need an account with unlimited CPU time.''

-- Landon Curt Noll, 1977
(on a CSUH data center form justifying the need for removal of CPU
time limits on Dr. Jurca's Cyber 174 account)