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119
Existence of primitive divisors of Lucas and Lehmer numbers
 J. Reine Angew. Math
, 2001
"... We prove that for n ? 30, every nth Lucas and Lehmer number has a primitive divisor. This allows us to list all Lucas and Lehmer numbers without a primitive divisor. Whether the mathematicians like it or not, the computer is here to stay. ..."
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Cited by 41 (0 self)
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We prove that for n ? 30, every nth Lucas and Lehmer number has a primitive divisor. This allows us to list all Lucas and Lehmer numbers without a primitive divisor. Whether the mathematicians like it or not, the computer is here to stay.
Information on combinatorial interpretation of Fibonomial coefficients
, 2004
"... A classicallike combinatorial interpretation of the Fibonomial coefficients is provided following [1,2]. An adequate combinatorial interpretation of recurrence satisfied by Fibonomial coefficients is also proposed. It is considered to be in the spirit classical combinatorial interpretation like b ..."
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Cited by 20 (15 self)
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A classicallike combinatorial interpretation of the Fibonomial coefficients is provided following [1,2]. An adequate combinatorial interpretation of recurrence satisfied by Fibonomial coefficients is also proposed. It is considered to be in the spirit classical combinatorial interpretation like binomial Newton and Gauss qbinomial coefficients or Stirling number of both kinds are. (See ref. [3,4] and refs. given therein). It also concerns choices. Choices of specific subsets of maximal chains from a nontree poset specifically obtained starting from the Fibonacci rabbits‘ tree. Several figures illustrate the exposition of statements the derivation of the recurrence itself included.
SumCracker: A package for manipulating symbolic sums and related objects
 J. Symb. Comput
"... We describe a new software package, named SumCracker, for proving and finding identities involving symbolic sums and related objects. SumCracker is applicable to a wide range of expressions for many of which there has not been any software available up to now. The purpose of this paper is to illustr ..."
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Cited by 19 (6 self)
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We describe a new software package, named SumCracker, for proving and finding identities involving symbolic sums and related objects. SumCracker is applicable to a wide range of expressions for many of which there has not been any software available up to now. The purpose of this paper is to illustrate how to solve problems using that package.
GENERATING FUNCTIONS FOR PRODUCTS OF POWERS OF FIBONACCI NUMBERS*
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Factorization of the tenth and eleventh Fermat numbers
, 1996
"... . We describe the complete factorization of the tenth and eleventh Fermat numbers. The tenth Fermat number is a product of four prime factors with 8, 10, 40 and 252 decimal digits. The eleventh Fermat number is a product of five prime factors with 6, 6, 21, 22 and 564 decimal digits. We also note a ..."
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Cited by 17 (8 self)
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. We describe the complete factorization of the tenth and eleventh Fermat numbers. The tenth Fermat number is a product of four prime factors with 8, 10, 40 and 252 decimal digits. The eleventh Fermat number is a product of five prime factors with 6, 6, 21, 22 and 564 decimal digits. We also note a new 27decimal digit factor of the thirteenth Fermat number. This number has four known prime factors and a 2391decimal digit composite factor. All the new factors reported here were found by the elliptic curve method (ECM). The 40digit factor of the tenth Fermat number was found after about 140 Mflopyears of computation. We discuss aspects of the practical implementation of ECM, including the use of specialpurpose hardware, and note several other large factors found recently by ECM. 1. Introduction For a nonnegative integer n, the nth Fermat number is F n = 2 2 n + 1. It is known that F n is prime for 0 n 4, and composite for 5 n 23. Also, for n 2, the factors of F n are of th...
Using Primitive Subgroups to Do More with Fewer Bits
, 2004
"... This paper gives a survey of some ways to improve the ef ciency of discrete logbased cryptography by using the restriction of scalars and the geometry and arithmetic of algebraic tori and abelian varieties. ..."
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Cited by 13 (3 self)
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This paper gives a survey of some ways to improve the ef ciency of discrete logbased cryptography by using the restriction of scalars and the geometry and arithmetic of algebraic tori and abelian varieties.
On the Iteration of Certain Quadratic Maps over GF(p)
"... We consider the properties of certain graphs based on iteration of the quadratic maps x ! x and x ! x 2 over a finite field GF(p). ..."
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Cited by 9 (0 self)
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We consider the properties of certain graphs based on iteration of the quadratic maps x ! x and x ! x 2 over a finite field GF(p).
On reciprocal sums of Chebyshev related sequences, Fibonacci Quart
 AMS Classification Numbers: 11B39, 11B37
, 1995
"... Define the sequences {Un}™=0 and {^}^Lo f ° r an Y r e a ' number/? by \Un = pUn_x + U„_2, UQ = 0, Ux = 1, n> 2, K = PV^+V^2, V0 = 2, Vx = p,n>2. (1.1) ..."
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Cited by 9 (0 self)
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Define the sequences {Un}™=0 and {^}^Lo f ° r an Y r e a ' number/? by \Un = pUn_x + U„_2, UQ = 0, Ux = 1, n> 2, K = PV^+V^2, V0 = 2, Vx = p,n>2. (1.1)
Dynamical Properties Of The Pascal Adic Transformation
 TH. DYN. SYS
, 2003
"... We study the dynamics of a transformation that acts on infinite paths in the graph associated with Pascal's triangle. For each ergodic invariant measure the asymptotic law of the return time to cylinders is given by a step function. We construct a representation of the system by a subshift on ..."
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Cited by 8 (5 self)
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We study the dynamics of a transformation that acts on infinite paths in the graph associated with Pascal's triangle. For each ergodic invariant measure the asymptotic law of the return time to cylinders is given by a step function. We construct a representation of the system by a subshift on a twosymbol alphabet and then prove that the complexity function of this subshift is asymptotic to a cubic, the frequencies of occurrence of blocks behave in a regular manner, and the subshift is topologically weak mixing.
The Divisibility Properties of Primary Lucas Recurrences with Respect to Primes." The Fibonacci Quarterly 18.4
"... In this paper we will extend the results of D. D. Wall [12], John Vinson [11]9 D. W. Robinson [9], and John H. Halton [3] concerning the divisibility properties of the Fibonacci sequence to the general Lucas sequence (r » _ r » ) / ( r i _ p 2). ..."
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Cited by 7 (2 self)
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In this paper we will extend the results of D. D. Wall [12], John Vinson [11]9 D. W. Robinson [9], and John H. Halton [3] concerning the divisibility properties of the Fibonacci sequence to the general Lucas sequence (r » _ r » ) / ( r i _ p 2).