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-regularity of the -adic valuation of the Fibonacci sequence
We show that the -adic valuation of the sequence of Fibonacci numbers is a
-regular sequence for every prime . For , we determine that
the rank of this sequence is , where is the
restricted period length of the Fibonacci sequence modulo .Comment: 7 pages; publication versio
The terms in Lucas sequences divisible by their indices
For Lucas sequences of the first kind (u_n) and second kind (v_n) defined as
usual for positive n by u_n=(a^n-b^n)/(a-b), v_n=a^n+b^n, where a and b are
either integers or conjugate quadratic integers, we describe the set of indices
n for which n divides u_n and also the set of indices n for which n divides
v_n. Building on earlier work, particularly that of Somer, we show that the
numbers in these sets can be written as a product of a so-called 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
Perfect powers in products of terms of elliptic divisibility sequences
Diophantine problems involving recurrence sequences have a long history and
is an actively studied topic within number theory. In this paper, we connect to
the field by considering the equation \begin{align*} B_mB_{m+d}\dots
B_{m+(k-1)d}=y^\ell \end{align*} in positive integers with
and , where is a fixed integer and
is an elliptic divisibility sequence, an important class
of non-linear recurrences. We prove that the above equation admits only
finitely many solutions. In fact, we present an algorithm to find all possible
solutions, provided that the set of -th powers in is given. (Note
that this set is known to be finite.) We illustrate our method by an example.Comment: To appear in Bulletin of Australian Math Societ
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