Concatenations of distinct repdigits in various linear recurrence sequencesMPIM
by
MPIM, Vivatsgasse, 7 - Lecture Hall
Max Planck Institute for Mathematics
Number theory lunch seminar
A repdigit is a natural number composed of repeated instances of the same digit in its decimal expansion, such as $11$, $555$, or $888888$. A palindrome is a natural number that reads the same forwards and backward, such as $121$, $3553$, or $9009$. Given a set of positive integers $ U $, one can ask about how many positive integers that are (palindromic) concatenations of distinct repdits belong to $ U $? This question leads to studying special forms of exponential Diophantine equations involving terms of linear recurrence sequences and concatenations of distinct repdigits. In this talk, I will survey recent results about this problem when $ U $ is the set of Perrin numbers, Tribonacci numbers, Padovan numbers, Narayana numbers, and Tribonacci-Lucas numbers. The proofs of these results heavily employ Baker's theory for nonzero lower bounds for linear forms in logarithms of algebraic numbers, reduction techniques involving the theory of continued fractions, and the reduced basis LLL algorithm. These results have been obtained in joint work with various colleagues such as H. Batte, T. P. Chalebgwa, P. Emong, F. Luca, and G. I. Mirumbe.