The aim of this paper is to describe the theory and implementation of the Elliptic Curve Primality Proving algorithm.

Consider lattice paths in Z 2 with three step types: the up diagonal (1; 1), the down diagonal (1; \Gamma1), and the double horizontal (2; 0). For n 1, let S n denote the set of such paths running from (0; 0) to (2n; 0) and remaining strictly above the x-axis except initially and terminally. It is well known that the cardinalities, r n = jS n j, are the large Schroder numbers. We use lattice paths to interpret bijectively the recurrence (n + 1)r n+1 = 3(2n \Gamma 1)r n \Gamma (n \Gamma 2)r n\Gamma1 , for n 2, with r 1 = 1 and r 2 = 2. We then use the bijective scheme to prove a result of Kreweras that the sum of the areas of the regions lying under the paths of S n and above the x-axis, denoted by AS n , satisfies AS n+1 = 6AS n \Gamma AS n\Gamma1 ; for n 2, with AS 1 = 1, and AS 2 = 7. Hence AS n = 1; 7; 41; 239; 1393; : : :. The bijective scheme yields analogous recurrences for elevated Catalan paths. Mathematical Reviews Subject Classification: 05A15 1 The paths and the recurr...

. We describe a primality testing algorithm, due essentially to Atkin, that uses elliptic curves over finite fields and the theory of complex multiplication. In particular, we explain how the use of class fields and genus fields can speed up certain phases of the algorithm. We sketch the actual implementation of this test and its use on testing large primes, the records being two numbers of more than 550 decimal digits. Finally, we give a precise answer to the question of the reliability of our computations, providing a certificate of primality for a prime number. IMPLEMENTATION DU TEST DE PRIMALITE D' ATKIN, GOLDWASSER, ET KILIAN R'esum'e. Nous d'ecrivons un algorithme de primalit'e, principalement du `a Atkin, qui utilise les propri'et'es des courbes elliptiques sur les corps finis et la th'eorie de la multiplication complexe. En particulier, nous expliquons comment l'utilisation du corps de classe et du corps de genre permet d'acc'el'erer les calculs. Nous esquissons l'impl'ementati...

The integral group ring of a finite group determines the isomorphism type of the chief factors of the group. Two proofs are given, one of which applies Cameron's and Teague's generalisation of Artin's theorem on the orders of finite simple groups to the orders of characteristically simple groups. The generalisation states that a direct power of a finite simple group is determined by its order with the same two types of exception which Artin found. Its proof, given here in detail, adapts and makes explicit certain functions of a natural number variable which Artin used implicitly. These functions contribute to the argument through a series of tables which supply their values for the orders of finite simple groups. 1.

An elevated Schroder path is a lattice path that uses the steps (1, 1), (1, -1), and (2, 0), that begins and ends on the x-axis, and that remains strictly above the x-axis otherwise. The total area of elevated Schroder paths of length 2n + 2 satisfies the recurrence f n+1 =6f n-f n-1 ,n#2, with the initial conditions f 0 =1,f 1 =7. A combinatorial interpretation of this recurrence is given, by first introducing sets of unrestricted paths whose cardinality also satisfies the recurrence relation and then establishing a bijection between the set of these paths and the set of triangles constituting the total area of elevated Schroder paths. 1 Introduction In the plane ZZ ZZ, we will use lattice paths with three steps types: a rise step defined by (1, 1), a fall step defined by (1, -1), and a horizontal step defined by (2, 0). A Schroder path is a sequence of rise, fall and horizontal steps running from (0, 0) to (2n,0) and remaining weakly above the x--axis. These paths are coun...

. We point out that a natural and direct way of analyzing the recurrence fm+1 = b m fm \Gamma fm\Gamma1 is by means of continued fractions theory. Applications to exotic numeration systems and to games are given, and previous examinations of the special case fm+1 = 6fm \Gamma fm\Gamma1 are pointed out. Denote by Z 0 and Z + the set of nonnegative integers and positive integers respectively. We denote simple continued fractions in the form (1) ff = a 0 + 1 a 1 + 1 a 2 + 1 . . . := [a 0 ; a 1 ; a 2 ; : : : ] ; and its convergents by pn =q n = [a 0 ; : : : ; an ] (n 0). Putting p \Gamma1 = 1, q \Gamma1 = 0, p 0 = a 0 , q 0 = 1, we then have pn = an pn\Gamma1 + pn\Gamma2 , q n = an q n\Gamma1 + q n\Gamma2 (n 1). See e.g., [HaWr1989, Ch. 10], [Per1950]. All the partial quotients a i are positive integers. Let un stand for either pn or q n , with the understanding that in each formula, either all u i denote p i or all denote q i . c fl0000 American Mathematical Society ???-????/00...

Abstract not found

We point out that a natural and direct way of analyzing the recurrence fm+1 = bmfm \Gamma fm\Gamma1 is by means of continued fractions theory. Applications to exotic numeration systems and to games are given, and previous examinations of the special case fm+1 = 6fm \Gamma fm\Gamma1 are pointed out.