# Illegal prime

In March

2001, Phil Carmody discovered a

prime number whose

binary representation corresponds to the gzipped

C source code of a

computer program implementing the

DeCSS decryption scheme and which is hence illegal under the

Digital Millennium Copyright Act. However, this has never been tested in a court, and it is possible that the number itself would be found to be legal, but not the information on how to use it to view DVDs.

The existence of infinitely many such primes is guaranteed by Dirichlet's theorem.

Protest against the indictment of DeCSS author Jon Johansen and legislation prohibiting publication of DeCSS code took many forms. One of them was the representation of the illegal code in a form that had an *intrinsically archivable* quality. Since the bits making up a computer program also represent a number, the plan was for the number to have some special property that would make it archivable and publishable. The primality of a number is a fundamental property, one outside the scope of the law.

The large prime database of the prime pages records the top 20 primes of various special forms; one of them is proof of primality using the elliptic curve primality proving (ECPP) algorithm. Thus, if the number were large enough, and proved prime using ECPP, it would be published.

By exploitation of the fact that the gzip program ignores bytes after the end of a null terminated compressed file, a set of candidate primes was generated, each of which would result in the DeCSS C code when unzipped. Of these several were identified as probable prime using the open source program OpenPFGW, and one of them was proved prime using the ECPP algorithm implemented by the *Titanix* software. At the time of discovery, this 1401 digit number was the tenth largest prime found using ECPP.

Following this, Carmody also created an executable illegal prime.

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