202 research outputs found
Coincidences in generalized Lucas sequences
For an integer , let be the generalized
Lucas sequence which starts with ( terms) and each term
afterwards is the sum of the preceding terms. In this paper, we find all
the integers that appear in different generalized Lucas sequences; i.e., we
study the Diophantine equation in nonnegative integers
with . The proof of our main theorem uses lower
bounds for linear forms in logarithms of algebraic numbers and a version of the
Baker-Davenport reduction method. This paper is a continuation of the earlier
work [4].Comment: 14 page
Horadam sequences: A survey update and extension.
We give an update on work relating to Horadam sequences that are generated by a general linear recurrence formula of order two. This article extends a ο¬rst ever survey published in early 2013 in this Bulletin, and includes coverage of a new research area opened up in recent times.N/
Splitting Fields and Periods of Fibonacci Sequences Modulo Primes
We consider the period of a Fibonacci sequence modulo a prime and provide an accessible, motivated treatment of this classical topic using only ideas from linear and abstract algebra. Our methods extend to general recurrences with prime moduli and provide some new insights. And our treatment highlights a nice application of the use of splitting fields that might be suitable to present in an undergraduate course in abstract algebra or Galois theory
Artin's primitive root conjecture -a survey -
This is an expanded version of a write-up of a talk given in the fall of 2000
in Oberwolfach. A large part of it is intended to be understandable by
non-number theorists with a mathematical background. The talk covered some of
the history, results and ideas connected with Artin's celebrated primitive root
conjecture dating from 1927. In the update several new results established
after 2000 are also discussed.Comment: 87 pages, 512 references, to appear in Integer
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