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

When Ramanujan died in 1920, he left behind an incomplete, unpublished manuscript in two parts on the partition function p(n) and, in contemporary terminology, Ramanujan’s tau-function τ(n). The first part, beginning with the Roman numeral I, is written on 43 pages, with the last nine comprising material for insertion in the

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We explicitly compute all the slopes of the Hecke operator U2 acting on overconvergent 2-adic level 1 cusp forms of weight 0: the nth slope is 1 + 2v((3n)!/n!), where v denotes the 2-adic valuation. We formulate an explicit conjecture about what these slopes should be for weight k forms.

. 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...

and tau functions

Abstract. The rank of an overpartition pair is a generalization of Dyson’s rank of a partition. The purpose of this paper is to investigate the role that this statistic plays in the congruence properties of pp(n), the number of overpartition pairs of n. Some generating functions and identities involving this rank are also presented. 1.

. The implementation of the Atkin-Goldwasser-Kilian primality testing algorithm requires the construction of the Hilbert class field of an imaginary quadratic field. We describe the computation of a defining equation for this field in terms of Weber's class invariants. The polynomial we obtain, noted W(X), has a solvable Galois group. When this group is dihedral, we show how to express the roots of this polynomial in terms of radicals. We then use these expressions to solve the equation W(X) j 0 mod p, where p is a prime. 1 Hilbert polynomials 1.1 Weber's functions We first introduce some functions. Let z be any complex number and put q = exp(2ißz). Dedekind's j function is defined by [21, x24 p. 85] j(z) = j(q) = q 1=24 Y m1 (1 \Gamma q m ): (1) We can expand j as [21, x34 p. 112] j(q) = q 1=24 0 @ 1 + X n1 (\Gamma1) n (q n(3n\Gamma1)=2 + q n(3n+1)=2 ) 1 A : (2) The Weber's functions are [21, x34 p. 114] f(z) = e \Gammaiß=24 j( z+1 2 ) j(z) ; (3) f 1 (z) = j...

A partition of a positive integer n is any non-increasing sequence of positive integers whose sum is n. Let p(n) denote the number of partitions of n. (As usual, we adopt the convention that p(0) = 1 and p(α) = 0 if α ̸ ∈ N). Ramanujan’s famous congruences, which were proved by Atkin, Ramanujan and Watson [2, 3, 13], assert that if j is a positive integer,

Let p(n) denote the number of partitions of a non-negative integer n. A well-known conjecture asserts that every arithmetic progression contains infinitely many integers M for which p(M) is odd, as well as infinitely many integers N for which p(N) is even (see Subbarao [22]). From the works of various authors, this conjecture has been verified for every arithmetic progression with modulus t when t = 1; 2; 3; 4; 5; 10; 12; 16; and 40: Here we announce that there indeed are infinitely many integers N in every arithmetic progression for which p(N) is even; and that there are infinitely many integers M in every arithmetic progression for which p(M) is odd so long as there is at least one such M . In fact if there is such an M , then the smallest such M 10 10 t 7 . Using these results and a fair bit of machine computation, we have verified the conjecture for every arithmetic progression with modulus t 100;000.