The Prime Internet Eisenstein Search

The Prime Internet Eisenstein Search, PIES, is a long-term effort to discover prime numbers. PIES is trying to exploit a property of a small class of numbers previously overlooked by other mathematicians, called Generalized Eisenstein Fermat numbers. These numbers have the newly discovered property that they are quicker and easier to prove prime than are typical numbers. Also in their favor is the fact that they are exceptionally dense in primes, more so than the candidates in any other prime-hunting project.

The PIES Project is orchestrated by Phil Carmody, a British mathematician living and working in Finland. Phil is the mathematician who discovered, back in 2001, the first “illegal” prime. This prime number can be unpacked into the original source code for DeCSS, the software that decodes the DVD encryption scheme. He also has discovered a second prime number that actually can execute the code.

Figure 1. PIES Logo

Contributors to PIES come from the US, Canada, Finland, Germany, France and a couple of other places around the world, although it is a relatively small international project. In true Linux form, the project is based all on volunteer work, runs on a small budget, is international and produces real results. The goal is pure research and somewhat esoteric—the discovery of large prime numbers of a slightly unusual form.

Prime numbers are those numbers that can be divided by 1 and themselves only. The numbers 1 and 0 are not considered prime, and the number 2 is the only even prime number. Primes are a fundamental part of our numbering system, and the search for prime numbers has fascinated mathematicians for more than two millennia. Today, prime numbers are used for public-key encryption, and large prime number searches are computationally intensive. The world's largest primes all are archived at Professor Chris Caldwell's “Prime Pages”, hosted at the University of Tennessee at Martin. Prime Pages not only archives the world's largest primes, but it also is the world's most complete resource for information on prime numbers.

The simplest method of determining whether a number is prime was understood by the ancient Greeks. Simply divide the number by primes smaller than the square root of the number being tested. Doing so finds all factors of the number; if none are found, the number is prime. This works reasonably well if your numbers are small, but when they get large, you need to be a bit smarter about how you search, calculate and prove that the number is indeed prime. Finding what you believe to be a prime number is not enough. Mathematicians are required to provide proof.

Bernard Riemann gave a lecture in 1859 in which he proposed a way to count prime numbers as a general rule. Proving what is known as the Riemann Hypothesis was one of the great mathematical challenges of the last century, and it continues to be so this century as well. Trying to figure out how many primes are in a range and what the distribution looks like within that range is an active area of research that helps drive the search for prime numbers.

Prime numbers are a kind of backbone for our number system. The use of prime numbers is more than simple intellectual play for mathematicians. Once Ron Rivest and his colleagues figured out that prime numbers were the way to make Whitfield Diffie's idea of asymmetrical, or public-key, cryptography a reality, prime numbers became indispensable. The more security required, the larger the prime numbers have to be.

The mathematics behind the PIES Project is somewhat esoteric and is explained partly on the project home page. It shares some properties with other large-prime-hunting projects, namely that it is a cyclotomic form, that is a factor of a^b–1. Other cyclotomic forms are Mersennes (2^p–1) and Generalized Fermat Numbers (b⁽²ⁿ⁾+1). The PIES primes are the first of the cyclotomic forms that can be found in large sizes, in large quantities and quickly but are not explicitly of the form a^b–1 or a^b+1. This particular PIES form, Generalized Eisenstein Fermat numbers, was first looked at in-depth by English amateur mathematician Mike Oakes several years before PIES started. But, it was because of Phil Carmody's advances in sieving—that is, quick removal of obvious non-primes because they have small, easily found factors—and fast primality testing algorithms that it became practical to look at the larger numbers with which PIES currently is working. Cyclotomic numbers are what you get from evaluating cyclotomic polynomials. The nth cyclotomic polynomial is denoted by Phi(n), and its value at b is denoted by Phi(n,b). Mersennes are Phi(p,2), and Generalized Fermat Numbers (GFNs) are Phi(2ⁿ,b). The PIES Generalized Eisenstein Fermats are Phi(3*2ⁿ,b).

Dr David Broadhurst of the Open University has been watching the development of the PIES Project with interest, although he has not devoted any cycles to it. When asked for his opinion, he said:

Although Phil is serious about mathematics and his various projects, he does it all for fun. His somewhat unusual sense of humor can be seen on the PIES Project home page. He believes that PIES is the only distributed computing project with a project song, for example. As one might guess from how the project name doesn't quite seem to parse correctly, it is indeed a complete contrivance, done simply so that the project name was fun and the search could be “themed”. Each fixed value of n in Phi(3*2ⁿ,b) defines a band in which primes can be hunted as b varies. Phil calls the small n=13 range “cherries”, the n=14 range “peaches” and the recently started n=15 range “apples”. Only he and his girlfriend, Anna, who assists with the project's image, words and song lyrics, know what the upcoming ranges will be called.