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155
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 46 (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.
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 15 (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.
Combinatorial interpretations of binomial coefficient analogues related to Lucas sequences
, 2009
"... Let s and t be variables. Define polynomials {n} in s, t by {0} = 0, {1} = 1, and {n} = s {n − 1} + t {n − 2} for n ≥ 2. If s, t are integers then the corresponding sequence of integers is called a Lucas sequence. Define an analogue of the binomial coefficients by n ..."
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Cited by 13 (6 self)
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Let s and t be variables. Define polynomials {n} in s, t by {0} = 0, {1} = 1, and {n} = s {n − 1} + t {n − 2} for n ≥ 2. If s, t are integers then the corresponding sequence of integers is called a Lucas sequence. Define an analogue of the binomial coefficients by n
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 12 (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.
Mahonian pairs
, 2011
"... We introduce the notion of a Mahonian pair. Consider the set, P∗ , of all words having the positive integers as alphabet. Given ﬁnite subsets S, T ⊂ P∗ , we say that (S, T ) is a Mahonian pair if the distribution of the major index, maj, over S is the same as the distribution of the inversion number ..."
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Cited by 12 (4 self)
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We introduce the notion of a Mahonian pair. Consider the set, P∗ , of all words having the positive integers as alphabet. Given ﬁnite subsets S, T ⊂ P∗ , we say that (S, T ) is a Mahonian pair if the distribution of the major index, maj, over S is the same as the distribution of the inversion number, inv, over T . So the wellknown fact that maj and inv are equidistributed over the symmetric group, Sn , can be expressed by saying that (Sn , Sn ) is a Mahonian pair. We investigate various Mahonian pairs (S, T ) with S = T . Our principal tool is Foata’s fundamental bijection φ : P∗ → P∗ since it has the property that maj w = inv φ(w) for any word w. We consider various families of words associated with Catalan and Fibonacci numbers. We show
that, when restricted to words in {1, 2}∗ , φ transforms familiar statistics on words into natural statistics on integer partitions such as the size of the Durfee square. The RogersRamanujan identities, the Catalan triangle, and various q analogues also make an appearance. We generalize the deﬁnition of Mahonian pairs to inﬁnite sets and use this as a tool to connect a partition bijection of CorteelSavageVenkatraman with the GreeneKleitman decomposition of a Boolean algebra into symmetric chains. We close with comments about future work and open problems.
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 10 (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)