. 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...

Abstract. Define ψm to be the smallest strong pseudoprime to all the first m prime bases. If we know the exact value of ψm, we will have, for integers n<ψm, a deterministic efficient primality testing algorithm which is easy to implement. Thanks to Pomerance et al. and Jaeschke, the ψm are known for 1 ≤ m ≤ 8. Upper bounds for ψ9,ψ10 and ψ11 were first given by Jaeschke, and those for ψ10 and ψ11 were then sharpened by the first author in his previous paper (Math. Comp. 70 (2001), 863–872). In this paper, we first follow the first author’s previous work to use biquadratic residue characters and cubic residue characters as main tools to tabulate all strong pseudoprimes (spsp’s) n < 1024 to the first five or six prime bases, which have the form n = pq with p, q odd primes and q − 1= k(p−1),k =4/3, 5/2, 3/2, 6; then we tabulate all Carmichael numbers < 1020, to the first six prime bases up to 13, which have the form n = q1q2q3 with each prime factor qi ≡ 3 mod 4. There are in total 36 such Carmichael numbers, 12 numbers of which are also spsp’s to base 17; 5 numbers are spsp’s to bases 17 and 19; one number is an spsp to the first 11 prime bases up to 31. As a result the upper bounds for ψ9,ψ10 and ψ11 are lowered from 20- and 22-decimal-digit numbers to a 19-decimal-digit number: ψ9 ≤ ψ10 ≤ ψ11 ≤ Q11 = 3825 12305 65464 13051 (19 digits) = 149491 · 747451 · 34233211. We conjecture that ψ9 = ψ10 = ψ11 = 3825 12305 65464 13051, and give reasons to support this conjecture. The main idea for finding these Carmichael numbers is that we loop on the largest prime factor q3 and propose necessary conditions on n to be a strong pseudoprime to the first 5 prime bases. Comparisons of effectiveness with Arnault’s, Bleichenbacher’s, Jaeschke’s, and Pinch’s methods for finding (Carmichael) numbers with three prime factors, which are strong pseudoprimes to the first several prime bases, are given. 1.

Introduction L'apparition des syst`emes de chiffrement `a clefs publiques de fa¸con g'en'erale [DH76], et du syst`eme de chiffrement RSA en particulier [ARS78], a caus'e un regain d'int'eret pour la th'eorie des nombres et en particulier l'arithm'etique dans ses aspects calculatoires. Pour r'epondre `a des questions aussi simples que celles concernant la d'ecomposition des nombres en facteurs premiers, il a fallu donner des r'eponses algorithmiques prenant en compte la faisabilit'e des calculs ainsi que le temps imparti pour donner une r'eponse satisfaisante. Cela a provoqu'e l'essor de la th'eorie algorithmique des nombres. Cet expos'e est destin'e `a mettre en lumi`ere les progr`es accomplis depuis une dizaine d'ann'ees dans les domaines de la primalit'e des entiers (comment peut-on prouver qu'un entier de quelques centaines de chiffres d'ecimaux est premier) ; factorisation des entiers (quels sont les facteurs d'un nombre qui n'est pas premier) ; logarithme

Fermat numbers are a classical topic in elementary number theory. Fermat introduced them and claimed that all these numbers are prime. This claim was disproofed by Euler who gave a property on the eventual divisors of the Fermat numbers. In this article we exhibit another serie whose definition is close to the one of Fermat numbers and which exhibit similar properties. This problem will lead us to the sets of covering congruences for numbers 2h:3 n + 1 as similarly Fermat numbers lead to Sierpinski's problem.

Fermat numbers are a classical topic in elementary number theory. Fermat introduced them and claimed that all these numbers are prime. This claim was disproofed by Euler who gave a property on the eventual divisors of the Fermat numbers. In this article we exhibit another serie whose definition is close to the one of Fermat numbers and which exhibit similar properties. This problem will lead us to the sets of covering congruences for numbers 2h:3 n + 1 as similarly Fermat numbers lead to Sierpinski's problem.