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12
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 167 (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.
Bijective Recurrences concerning Schröder paths
, 1998
"... 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 xaxis except initially and terminally. It ..."
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Cited by 11 (1 self)
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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 xaxis 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 xaxis, 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...
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 implem ..."
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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.
Composition factors from the group ring and Artin's theorem on orders of simple groups
 Proc. London Math. Soc
, 1990
"... 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 si ..."
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Cited by 8 (2 self)
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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.
A Combinatorial Interpretation of the Area of Schröder Paths
 Electronic J. Combinatorics
, 1999
"... 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 xaxis, and that remains strictly above the xaxis otherwise. The total area of elevated Schroder paths of length 2n + 2 satisfies the recurrence f n+1 =6f nf n1 ,n#2, with ..."
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Cited by 6 (2 self)
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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 xaxis, and that remains strictly above the xaxis otherwise. The total area of elevated Schroder paths of length 2n + 2 satisfies the recurrence f n+1 =6f nf n1 ,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 xaxis. These paths are coun...
On the Recurrence . . .
"... 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 ou ..."
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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.
ON THE INFINITUDE OF COMPOSITE NSW NUMBERS
, 2000
"... approximately 20 years ago in connection with the order of certain simple groups. These are the numbers fn which satisfy the recurrence / » i = 6/w/*i 0) with initial conditions fi = l and f2 = 7. These numbers have also been studied in other contexts. For example, Bonk, Shapiro, and Simion [2] d ..."
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approximately 20 years ago in connection with the order of certain simple groups. These are the numbers fn which satisfy the recurrence / » i = 6/w/*i 0) with initial conditions fi = l and f2 = 7. These numbers have also been studied in other contexts. For example, Bonk, Shapiro, and Simion [2] discuss them in relation to Schroder numbers and combinatorial statistics on lattice paths. Recently, Barcucci et al. [1] provided a combinatorial interpretation for the NSW numbers by defining a certain regular language 2J and studying particular properties of 2J. They close their note by asking two questions: 1. Do there exist infinitely many fn prime? 2. Do there exist infinitely many fn composite? The goal of this paper is to answer the second question affirmatively, but in a much broader context. Fix an integer k>2 and consider the sequence of values satisfying fn+l = kfn~~fn\> fi = l9 and f2 = k +1. Then we have the following theorem.
On the Recurrence F<sub>m+1</sub> = B<sub>m</sub>f<sub>m</sub>F<sub>m1</sub> and Applications
"... . 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 ..."
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. 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...
and
, 2006
"... In a recent note, Santana and Diaz–Barrero proved a number of sum identities involving the well–known Pell numbers. Their proofs relied heavily on the Binet formula for the Pell numbers. Our goal in this note is to reconsider these identities from a purely combinatorial viewpoint. We provide biject ..."
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In a recent note, Santana and Diaz–Barrero proved a number of sum identities involving the well–known Pell numbers. Their proofs relied heavily on the Binet formula for the Pell numbers. Our goal in this note is to reconsider these identities from a purely combinatorial viewpoint. We provide bijective proofs for each of the results by interpreting the Pell numbers as enumerators of certain types of tilings. In turn, our proofs provide helpful insight for straightforward generalizations of a number of the identities.