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Elliptic Curves And Primality Proving
 Math. Comp
, 1993
"... The aim of this paper is to describe the theory and implementation of the Elliptic Curve Primality Proving algorithm. ..."
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Cited by 162 (22 self)
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The aim of this paper is to describe the theory and implementation of the Elliptic Curve Primality Proving algorithm.
Ramanujan’s unpublished manuscript on the partition and tau functions with proofs and commentary
 Sém. Lotharingien de Combinatoire 42 (1999), 63 pp.; in The Andrews Festschrift
, 2001
"... When Ramanujan died in 1920, he left behind an incomplete, unpublished manuscript in two parts on the partition function p(n) and, in contemporary terminology, Ramanujan’s taufunction τ(n). The first part, beginning with the Roman numeral I, is written on 43 pages, with the last nine comprising mat ..."
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Cited by 26 (13 self)
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When Ramanujan died in 1920, he left behind an incomplete, unpublished manuscript in two parts on the partition function p(n) and, in contemporary terminology, Ramanujan’s taufunction τ(n).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
Distribution Of The Partition Function Modulo Composite Integers M
 M, MATH. ANNALEN
, 2000
"... ..."
Slopes of overconvergent 2adic modular forms
 COMPOSITIO MATH.
, 2005
"... We explicitly compute all the slopes of the Hecke operator U2 acting on overconvergent 2adic level 1 cusp forms of weight 0: the nth slope is 1 + 2v((3n)!/n!), where v denotes the 2adic valuation. We formulate an explicit conjecture about what these slopes should be for weight k forms. ..."
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Cited by 12 (2 self)
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We explicitly compute all the slopes of the Hecke operator U2 acting on overconvergent 2adic level 1 cusp forms of weight 0: the nth slope is 1 + 2v((3n)!/n!), where v denotes the 2adic valuation. We formulate an explicit conjecture about what these slopes should be for weight k forms.
Implementation Of The AtkinGoldwasserKilian Primality Testing Algorithm
 Rapport de Recherche 911, INRIA, Octobre
, 1988
"... . 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 impl ..."
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Cited by 9 (7 self)
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. 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...
On some claims in Ramanujan’s ‘unpublished’ manuscript on the partition and tau functions
 The Ramanujan Journal. HEXAGONAL VERSUS THE SQUARE LATTICE 473
"... and tau functions ..."
Rank and congruences for overpartition pairs
 Int. J. Number Theory
"... 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 invol ..."
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Cited by 7 (4 self)
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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.
Extension of Ramanujan’s congruences for the partition function modulo powers of 5
 J. Reine Angew. Math
"... A partition of a positive integer n is any nonincreasing 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 an ..."
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Cited by 6 (4 self)
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A partition of a positive integer n is any nonincreasing 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,
Construction Of Hilbert Class Fields Of Imaginary Quadratic Fields And Dihedral Equations Modulo p
, 1989
"... . The implementation of the AtkinGoldwasserKilian 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, note ..."
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Cited by 4 (3 self)
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. The implementation of the AtkinGoldwasserKilian 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...
Quadratic twists and the coefficients of weakly holomorphic modular forms
 J. Ramanujan Math. Soc
"... Abstract. Linear congruences have been shown to exist for the partition function and for several other arithmetic functions that are generated by weakly holomorphic modular forms. In previous work, the author showed that such congruences exist for the Fourier coefficients of any weakly holomorphic m ..."
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Cited by 4 (1 self)
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Abstract. Linear congruences have been shown to exist for the partition function and for several other arithmetic functions that are generated by weakly holomorphic modular forms. In previous work, the author showed that such congruences exist for the Fourier coefficients of any weakly holomorphic modular form in a large class. In this paper, quadratic twists are used to produce new congruences for the coefficients of these forms. To demonstrate our main theorems, we give new congruences for overpartitions, and extend a result of Penniston regarding congruences for ℓregular partitions.