Overview

Mersenne numbers are of the simple form 2^P-1, where P is the exponent to be tested. It is easy to prove that if 2^P-1 is prime, then P must be a prime. Thus, the first step in the search is to create a list of prime exponents to test.

If a Mersenne number has a factor it cannot be a Mersenne Prime, so the first step is to spend some time trying various factorization methods to find small factors to eliminate candidate exponents. The amount of effort spent looking for factors is balanced against the probability of finding a factor. There are two main factoring tests that GIMPS uses: trial factoring (TF) and P-1 (see detailed explanations on the Math Page if interested).

If no factor is found after an appropriate amount of effort, the Mersenne number is tested with a PRP (Probable Prime) test. If that says composite, the candidate is not a Mersenne Prime. In the (rare) case that PRP says probably prime, the number is re-tested with Lucas-Lehmer test to verify primality.

At this point the Mersenne number is known to be either prime or composite, but for some of the smallest composite Mersenne numbers an additional level of factoring is attempted using the Elliptic Curve Method (ECM) which may be able to find some factors too large for TF or P-1.

In brief, you need:

1. A relatively modern computer. Older computers can still be useful to help other computers find primes.
2. Your computer on and running more often than off, at least most of the time; it's OK to go on holidays, vacations, etc.
3. Time and patience. Testing for a prime can take weeks even on the fastest computers and months on very old computers.
4. An Internet connection. Testing on offline computers is possible, but beyond the scope of this article.

Most importantly, you will need a lot of patience. Roughly speaking it will take about a month to run a single primality test.

If you've decided you have a powerful enough computer and enough patience to stick with the project, then visit the download page for instructions on downloading, installing, and starting the program.

Prime95 talks to PrimeNet, a central server on the Internet, to get work to do and report results. The program communicates using the HTTP protocol and can be configured with proxy information to pass through common firewalls. The program does not require a continuous Internet connection, data is sent to/from the server approximately once an hour. Most work types require just a few bytes of data to be sent, PRP results will transmit several hundred megabytes on completion of the PRP test (typically this would be about 1GB per month total).

GPU software communicates with PrimeNet using AutoPrimeNet. See the Quick Start Guide for how to set that up if applicable.

There are several types of work assigned by the server (see glossary for definitions). Users can choose which type of work best suits their computer:

• Most users will do PRP tests to have a chance at finding the next Mersenne Prime.
• If you have a system with a large amount of RAM (64GB+) that is mostly unused, P-1 factoring would be a good choice.
• Older systems that don't have much RAM or disk space are well suited to LL-DC work, double-checking old unconfirmed tests (there's still a small chance a previous test was wrong and there's a Prime waiting to be discovered).
• If you have an NVIDIA or AMD GPU in your system, you can also run other tests simultaneously to those on your CPU. NVIDIA GPUs are best suited to trial factoring work with mfaktc, AMD GPUs are best suited to PRP work with PRPLL.

Every half hour the program saves its state so that if there is a computer crash you will lose at most a half-hour of work. All temporary files and program settings are conveniently stored in the same folder as the program.

Before you complete your assignment, the program will get more work to do from PrimeNet. This will assure that your computer has a continuous supply of work. When the work assignment is completed the program returns one of several possible types of results to PrimeNet, then proceeds with the next work assignment task.

If you are lucky enough to find a new Mersenne prime, the program notifies the server and optionally emits a continuous sound to notify you of the happy news! There may be a cash research discovery award for your new prime!