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...

The Cunningham project seeks to factor numbers of the form b n \Sigma 1 with b = 2; 3; : : : small. One of the most useful techniques is Aurifeuillian Factorization whereby such a number is partially factored by replacing b by a polynomial in such a way that polynomial factorization is possible. For example, by substituting y = 2 k into the polynomial factorization (2y 2 ) 2 + 1 = (2y 2 \Gamma 2y + 1)(2y 2 + 2y + 1) we can partially factor 2 4k+2 + 1. Schinzel [Sch] gave a list of such identities that have proved useful in the Cunningham project; we believe that Schinzel identified all numbers that can be factored by such identities and we prove this if one accepts our definition of what "such an identity" is. We then develop our theme to similarly factor f(b n ) for any given polynomial f , using deep results of Faltings from algebraic geometry and Fried from the classification of finite simple groups.

. There are 38975 Fermat pseudoprimes (base 2) up to 10 11 , 101629 up to 10 12 and 264239 up to 10 13 : we describe the calculations and give some statistics. The numbers were generated by a variety of strategies, the most important being a back-tracking search for possible prime factorisations, and the computations checked by a sieving technique. 1 Introduction A (Fermat) pseudoprime (base 2) is a composite number N with the property that 2 N \Gamma1 j 1 mod N . For background on pseudoprimes and primality tests in general we refer to Bressoud [1], Brillhart et al [2], Koblitz [4], Ribenboim [12] and [13] or Riesel [14]. Previous tables of pseudoprimes were computed by Pomerance, Selfridge and Wagstaff [11]. We have shown that there are 38975 pseudoprimes up to 10 11 , 101629 up to 10 12 and 264239 up to 10 13 ; all have at most 9 prime factors. Let P (X) denote the number of pseudoprimes less than X and let P (d; X) denote the number with exactly d prime factors. In ...