Great Internet Mersenne Prime Search
Great Internet Mersenne Prime Search
Great Internet Mersenne Prime Search
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<strong>Great</strong> <strong>Internet</strong><br />
<strong>Mersenne</strong> <strong>Prime</strong> <strong>Search</strong><br />
(GIMPS)<br />
Stephen Ratzlaff
What is a <strong>Mersenne</strong> <strong>Prime</strong><br />
• If M n = 2 n -1 is prime then it is a <strong>Mersenne</strong><br />
<strong>Prime</strong><br />
• If 2 n -1 is prime then n must be prime<br />
o I won't show you the proof<br />
o So it is usually written as M P = 2 P -1
GIMPS<br />
• <strong>Search</strong>es for <strong>Mersenne</strong> prime numbers<br />
• Distributed computing project<br />
o Central server hands out work to volunteers<br />
o Volunteers calculate primality of numbers<br />
o Volunteers report back to server their results
Volunteers<br />
• Theory is that most people do not use<br />
their computer's to the fullest all the time.<br />
• They can volunteer their spare resources<br />
to help the community<br />
o Low priority process<br />
o A little extra power<br />
o Low impact to user<br />
o Huge benefit to community
<strong>Prime</strong>Net<br />
• Network of volunteers and server that run<br />
GIMPS<br />
• ~130 TFLOP/s<br />
o Current 30 day average
<strong>Prime</strong>Net Server<br />
• Distributes work to clients<br />
o Primality Tests<br />
Faster clients<br />
o Test Candidates<br />
Slower clients<br />
• Collects results from clients
Work Assignments<br />
• Elimination by factoring<br />
o M P can be factored then it is not a<br />
prime<br />
• Lucas-Lehmer primality test<br />
o Very expensive test<br />
Test for 1 prime is currently estimated at 45 days<br />
on my PC
Elimination by Factoring<br />
• Slower computers are given the assigned<br />
these jobs<br />
• This can eliminate candidates quicker<br />
than if done by the Lucas-Lehmer test<br />
• If it passes x amount of iterations then the<br />
candidate gets passed onto the Lucas-<br />
Lehmer test
Lucas-Lehmer Primality Test<br />
• Test for <strong>Mersenne</strong> primality<br />
• Assigned to faster computers<br />
// Determine if Mp = 2p − 1 is prime<br />
Lucas–Lehmer(p)<br />
var s = 4<br />
var M = 2p − 1<br />
repeat p − 2 times:<br />
s = ((s × s) − 2) mod M<br />
if s = 0 return PRIME else return COMPOSITE
Double Checking<br />
• Because floating points are used there is<br />
some error<br />
• After the first round check is done a<br />
double check is assigned by the <strong>Prime</strong>Net<br />
Server<br />
• 1.5% error rate
Results<br />
• Factoring Test<br />
o Not prime, factor is recorded<br />
o No factors, needs more testing<br />
• Lucas-Lehmer Test<br />
o Not prime<br />
o <strong>Prime</strong>
It's a pretty cool concept<br />
• Central server hands out work<br />
assignments<br />
• A bunch of volunteers work on small<br />
problems<br />
• As seen a lot of raw processing power<br />
can be harnessed through this method.
Demo
Notes<br />
• GIMPS Homepage<br />
o http://www.mersenne.org/<br />
• M P = 2 P -1 proof<br />
o http://primes.utm.edu/notes/proofs/Theorem2<br />
.html<br />
• Lucas Lehmer Proof<br />
o http://primes.utm.edu/notes/proofs/LucasLeh<br />
mer.html
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