this paper appeared in "Design and Implementation of Symbolic Computation Systems," A. Miola (ed.), Springer Lect. Notes Comput. Science, 722, 66--80 (1993).

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...

. Rabin's algorithm is commonly used in computer algebra systems and elsewhere for primality testing. This paper presents an experience with this in the Axiom* computer algebra system. As a result of this experience, we suggest certain strengthenings of the algorithm. Introduction It is customary in computer algebra to use the algorithm presented by Rabin [1980] to determine if numbers are prime (and primes are needed throughout algebraic algorithms). As is well known, a single iteration of Rabin's algorithm, applied to the number N , has probability at most 0.25 of reporting "N is probably prime", when in fact N is composite. For most N , the probability is much less than 0.25. Here, "probability" refers to the fact that Rabin's algorithm begins with the choice of a "random" seed x, not congruent to 0 modulo N . In practice, however, true randomness is hard to achieve, and computer algebra systems often use a fixed set of x --- for example Axiom release 1 uses the set f3; 5; 7; 11;...

We make an attempt to compare the speed of eeme primality testing algorithms for certifying loo-digit prime numbers. 1. Introduction. The