56 research outputs found
On -coordinates of Pell equations which are repdigits
Let be a given integer. In this paper, we show that there only
finitely many positive integers which are not squares, such that the Pell
equation has two positive integer solutions with the
property that their -coordinates are base -repdigits. Recall that a base
-repdigit is a positive integer all whose digits have the same value when
written in base . We also give an upper bound on the largest such in
terms of .Comment: To appear in The Fibonacci Quarterly Journa
k-generalized Fibonacci numbers which are concatenations of two repdigits
We show that the k-generalized Fibonacci numbers that are concatenations of two repdigits have at most four digits
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
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