We describe the primality testing algorithms in use in some popular computer algebra systems, and give some examples where they break down in practice. 1 Introduction In recent years, fast primality testing algorithms have been a popular subject of research and some of the modern methods are now incorporated in computer algebra systems (CAS) as standard. In this review I give some details of the implementations of these algorithms and a number of examples where the algorithms prove inadequate. The algebra systems reviewed are Mathematica, Maple V, Axiom and Pari/GP. The versions we were able to use were Mathematica 2.1 for Sparc, copyright dates 1988-1992; Maple V Release 2, copyright dates 1981-1993; Axiom Release 1.2 (version of February 18, 1993); Pari/GP 1.37.3 (Sparc version, dated November 23, 1992). The tests were performed on Sparc workstations. Primality testing is a large and growing area of research. For further reading and comprehensive bibliographies, the interested re...

Note: This document is available in other formats. Some years ago I visited M. F. Perutz, the Cambridge biochemist who deciphered the structure of the hemoglobin molecule. Professor Perutz showed me a series of artifacts from his 20-year struggle to unravel the twists and folds of the oxygen-carrying protein. There was an x-ray diffraction film whose symmetrical array of dots looked like a lace doily, a contour map of electron density drawn on stacked sheets of transparent plastic, and a huge molecular model supported by a forest of brass rods. Looking at these objects, I could understand in a general way how an x-ray diffraction pattern reveals the structure of a crystallized molecule. The diffraction pattern is the Fourier transform of the crystal lattice, representing in "frequency space " the positions of the atoms in ordinary space. The lattice and the diffraction pattern have a reciprocal relationship: Widely separated dots on the x-ray film correspond to closely spaced planes of atoms in the crystal, and nearby dots on the film are generated by widely spaced atoms. That much I understood. What was lacking was more-specific knowledge of how to translate from crystal lattice to diffraction pattern and back again. I wanted to look at the x-ray film and see the geometry of the crystal. What I wanted most was a chance to experiment and explore. I wanted to nudge an atom in the crystal, and see how that displacement altered the diffraction pattern.

In this paper we develop a representational approach to media theory. We construct representations of media by well graded families of sets and partial cubes and establish the uniqueness of these representations. Two particular examples of media are also described in detail.