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Primitive divisors of elliptic divisibility sequences
 J. Number Theory
"... Abstract. Silverman proved the analogue of Zsigmondy’s Theorem for elliptic divisibility sequences. For elliptic curves in global minimal form, it seems likely this result is true in a uniform manner. We present such a result for certain infinite families of curves and points. Our methods allow the ..."
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Cited by 15 (7 self)
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Abstract. Silverman proved the analogue of Zsigmondy’s Theorem for elliptic divisibility sequences. For elliptic curves in global minimal form, it seems likely this result is true in a uniform manner. We present such a result for certain infinite families of curves and points. Our methods allow the first explicit examples of the elliptic Zsigmondy Theorem to be exhibited. 1.
The terms in Lucas sequences divisible by their indices. On arXiv: 0908.3832v1[math.NT
, 2009
"... Abstract. For Lucas sequences of the first kind (un)n≥0 and second kind (vn)n≥0 defined as usual by un = (α n − β n)/(α − β), vn = α n + β n, where α and β are either integers or conjugate quadratic integers, we describe the sets {n ∈ N: n divides un} and {n ∈ N: n divides vn}. Building on earlier w ..."
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Cited by 4 (2 self)
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Abstract. For Lucas sequences of the first kind (un)n≥0 and second kind (vn)n≥0 defined as usual by un = (α n − β n)/(α − β), vn = α n + β n, where α and β are either integers or conjugate quadratic integers, we describe the sets {n ∈ N: n divides un} and {n ∈ N: n divides vn}. Building on earlier work, particularly that of Somer, we show that the numbers in these sets can be written as a product of a socalled basic number, which can only be 1, 6 or 12, and particular primes, which are described explicitly. Some properties of the set of all primes that arise in this way is also given, for each kind of sequence. 1.
On the Brightness of the Thomson Lamp. A Prolegomenon to Quantum Recursion Theory
, 2009
"... Some physical aspects related to the limit operations of the Thomson lamp are discussed. Regardless of the formally unbounded and even infinite number of “steps” involved, the physical limit has an operational meaning in agreement with the Abel sums of infinite series. The formal analogies to accele ..."
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Cited by 1 (1 self)
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Some physical aspects related to the limit operations of the Thomson lamp are discussed. Regardless of the formally unbounded and even infinite number of “steps” involved, the physical limit has an operational meaning in agreement with the Abel sums of infinite series. The formal analogies to accelerated (hyper) computers and the recursion theoretic diagonal methods are discussed. As quantum information is not bound by the mutually exclusive states of classical bits, it allows a consistent representation of fixed point states of the diagonal operator. In an effort to reconstruct the selfcontradictory feature of diagonalization, a generalized diagonal method allowing no quantum fixed points is proposed.
SYNCHRONIZATION POINTS AND ASSOCIATED DYNAMICAL INVARIANTS
"... Abstract. This paper introduces new invariants for multiparameter dynamical systems. This is done by counting the number of points whose orbits intersect at time n under simultaneous iteration of finitely many endomorphisms. We call these points synchronization points. The resulting sequences of cou ..."
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Cited by 1 (1 self)
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Abstract. This paper introduces new invariants for multiparameter dynamical systems. This is done by counting the number of points whose orbits intersect at time n under simultaneous iteration of finitely many endomorphisms. We call these points synchronization points. The resulting sequences of counts together with generating functions and growth rates are subsequently investigated for homeomorphisms of compact metric spaces, toral automorphisms and compact abelian group epimorphisms. Synchronization points are also used to generate invariant measures and the distribution properties of these are analysed for the algebraic systems considered. Furthermore, these systems reveal strong connections between the new invariants and problems of active interest in number theory, relating to heights and greatest common divisors. 1.
Primitive Prime Divisors of FirstOrder Polynomial Recurrence Sequences
, 2006
"... The question of which terms of a recurrence sequence fail to have primitive prime divisors has been significantly studied for several classes of linear recurrence sequences and for elliptic divisibility sequences. In this paper, we consider the question for sequences generated by the iteration of a ..."
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The question of which terms of a recurrence sequence fail to have primitive prime divisors has been significantly studied for several classes of linear recurrence sequences and for elliptic divisibility sequences. In this paper, we consider the question for sequences generated by the iteration of a polynomial. For two classes of polynomials f(x) ∈ Z[x] and initial values a1 ∈ Z, we show that the sequence (an) given by an+1 = f(an) for n ≥ 1 has only finitely many terms which have no primitive prime divisor. 1 Introduction and Statement of Results Given a sequence (an) = (a1, a2,...) of integers, we say that a term an of the sequence has a primitive prime divisor if there exists a prime p such that pan, but p ∤ ai for i < n. For a given sequence (an), we can ask a natural question: which terms of the sequence have primitive prime divisors?
Combinatorial Functions
"... First and foremost, I am grateful to my Ph.D. advisor Prof. Janos Makowsky. Janos taught me how to do research and how to do mathematics, how to ask the right questions and how to answer them. I am grateful to him for allowing me the freedom to pursue my research as I saw fit, while at the same time ..."
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First and foremost, I am grateful to my Ph.D. advisor Prof. Janos Makowsky. Janos taught me how to do research and how to do mathematics, how to ask the right questions and how to answer them. I am grateful to him for allowing me the freedom to pursue my research as I saw fit, while at the same time being there to provide support and encouragement. His unique ability to see the big picture and his enthusiasm for doing research have been a great inspiration to me during my studies. I consider myself fortunate to have had him as my advisor. I am also grateful for the generous funding he provided me through his grants. I am indebt to my coauthors, Dr. Ilia Averbouch, Prof. Eldar Fischer, Benny Godlin, Dr. Elena Ravve and Prof. Boris Zilber. I am grateful to my Ph.D. examiners Prof. Amir Shpilka, Prof. Raphael Yuster and Prof. Shmuel Onn for useful comments on an earlier version of this thesis. I am thankful to Prof. Ira Gessel for hosting me at Brandeis University in the Summer of 2010. I am also thankful to many friends and fellow students with whom I consulted and shared ideas and coffee breaks. Last, but not least, I am grateful to my family for their constant support and their invaluable advice. This thesis is dedicated to them.