Reaching new heights with new Mersenne prime! (12/07/2001)  (Another of the Prime Pages' resources)
 New record prime: 277,232,917-1 with 23,249,425 digits by Pace, Woltman, Kurowski, Blosser & GIMPS (26 Dec 2017).
There are some places that few will ever go.  Perhaps the road is too long or the air is too thin.  Often the length of the trip alone is prohibitive.  Michael Cameron of Owen Sound, Ontario has just completed such a journey--he has found the largest known prime:
213466917-1

Now 20 years old, Michael is a student by day is training for a position with NuComm International, Inc. at night.  A few months ago Michael joined GIMPS (the Great Internet Mersenne Prime Search) and began to look for a new record Mersenne prime.  This is a daunting task!  In over 2000 years of searching, only 38 have been found.  Now counting Michael's, there are 39. Michael's new behemoth has 4,053,946 digits and would easily fill a paperback novel.

A Mersenne prime is a prime number that has the form 2p-1.  The ancient Greek geometer Euclid explained the connection between these primes and even perfect numbers in his classic geometry text "The Elements."  The French monk Marin Mersenne stirred up a great deal of interest in the middle ages with a wild claim about which of these numbers were prime.  Mathematicians continue to study the distribution of the primes and especially the distribution of Mersenne primes.

Was it hard?

In one sense what Michael did was very simple: he decided not to waste his computer's idle time.  As he explains:

"My friend informed me that I was wasting my CPU power and that if I was going to leave my computer on all the time I should make use of that wasted CPU time.  He recommended Prime95 but at first I was hesitant.  I wanted something that I could see happening.  I wanted instant results.  So I tried some other programs but they were not what I had expected.  I decided to put Prime95 on my system because running it is like running nothing at all."

Prime95 is a free program written by George Woltman and provided on his GIMPS site.   Just download the software, install it, and let it run.  Simple and easy, but the road is long.  To test a single number takes days on the fastest microcomputers, months on others.  Michael's "AMD T-Bird 800Mhz with 512 Megs of RAM"  took 42 days.

"I was so surprised about the beeping and thought that there was something wrong because I had forgotten all about it."

But in another sense, what Michael did has proven very difficult--he persisted.   Over 86% of those who have begun to test a Mersenne exponent with GIMPS have never completed even one, and the average GIMPS participant has completed less than three (mean 2.67).  This was Michael's fourth.

And Michael was also very lucky.  Mersenne primes appear to be scattered randomly.  Michael just hit the jack pot.  Others have tested 3000, 4000, and one person almost 8000 exponents, but it was Michael who found this one.  In this sense Michael is a newcomer: of those that have completed at least one exponent, the average is more than 19 (mean 19.54).

Not one succeeds here alone

Michael was able to succeed because George Woltman wrote the software.  Michael needed his help, so George properly will share credit for this (his fifth!) Mersenne prime.

Scott Kurowski, founder of Entropia, Inc., also deserves equal billing.  His company provides the GIMPS project with a free central server, and with the necessary components to make the whole process of virtually invisible to the users.  Because of Scott's work GIMPS participants now include over 130,000 people running more than 205,000 PC's in virtually every time zone in the world!  Scott's company is also involved in Aids research and economic research.

This is also a victory for all of the thousands involved in GIMPS. Perhaps the most important contribution of the project to mathematics is not the primes it finds, but showing that there are no other primes in between that have been missed (and this has happened before).  So each time a participant completes a test and find these number is not prime, they have actually done something quite positive!

The new Mersenne prime was independently verified by two others using programs on different operating systems:

• Guillermo Ballester Valor used his program Glucas v.2.8c on a Compaq server (Blazer Itanium with four Processors).  The operating system was Red Hat Linux 7.1.
• Paul Victor Novarese of Compaq Corp. used a program written by Ernst Mayer, on a 667 MHz Compaq Alpha workstation.  The calculation took about three weeks.

A Little Theory Never Hurts

Integers greater than one are called prime if they can not be factored (10 is not prime because it has the prime factors 2 and 5). To show a small number is prime, we can just divide by the primes below its square root. For example, 53 is prime because it is not divisible 2, 3, 5 or 7.

Large numbers need more complicated tests that usually involve a form of Lagrange's theorem. If you look at a list of the largest known primes you will see that for most such number n, either n+1 or n-1 factors easily. Mersenne primes follow this rule and are the primes of the form 2p-1. It is easy to show that p here must also be a prime. Because of the special form of the Mersennes, there is an easy test to see if they are prime, called the Lucas-Lehmer test.