April 19th 2006
# Great Internet Mersenne Prime Search - GIMPS

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## 1. Introduction

The Great Internet Mersenne Prime Search is one of the very first distributed search internet projects.
Since 1996 thousends of volunteers with tenthousends of CPU's are hunting for world record primes.
As of April 2006, GIMPS is holding the 8 largest Mersenne prime numbers and the 4 biggest primes at all.

A Mersenne prime is a prime of the form *M*_{p} = 2^{p} - 1. It is easy
to see that the exponent *p* necessarily has to be prime.
The residue of the Lucas-Lehmer Test gives
definite information whether *M*_{p} is prime or composite.
Factors of the Mersenne number *M*_{p} must have the form 2 *k* *p* + 1. The workflow choosen at the GIMPS project is:
- Try all possible factors 2
*k* *p* + 1 of *M*_{p} below a given border. For 10-million digit numbers this border is 69 Bits
and takes about 2 days.
Most numbers can be eliminated this way.
- Apply a classical 2-step Pollard p-1 Test with
appropriate bounds
*B*_{1} and *B*_{2}. The chance of finding a factor in about 2 days is above 5%.
- Perform the Lucas-Lehmer Test. This can take many weeks
and applies heavy usage of highly optimized Fast Fourier Transformations.

Personally, I am very proud to have been one of the very first participants in 1996 to join this project. In the late 90's I had up to
17 CPU's running while at the moment I try to keep an average of 2000 P90 CPU hrs/day on 3-4 CPU's. This gives me a rank of around 250 with
some 100 P90 CPU years.

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