Abstract
In this paper, we study the problem of the explicit intersection of two sequences. More specifically, we find all repdigits (i.e., numbers with only one repeated digit in its decimal expansion) which can be written as the product of a Fibonacci by a Tribonacci number (both with the same indexes). To work on this problem, our approach is to combine lower bounds from the Baker’s theory with reduction methods (based on the theory of continued fractions) due to Dujella and Pethö.
Highlights
Before starting with the main problem of this paper, we recall some nomenclature and symbols for the convenience of the reader: The Fibonacci sequence ( Fn )n is defined by the recurrenceFn+1 = Fn + Fn−1, (1)with initial values F0 = 0 and F1 = 1.The Tribonacci numbers t n ( Tn )n are defined by the third-order recurrenceTn+1 = Tn + Tn−1 + Tn−2, (2)with initial values T0 = 0 and T1 = T2 = 1.A repdigit is a number of the form a 10` − 1 ! (3)
We found all repdigits which can be written as a product of a Fibonacci number and a Tribonacci number
We combined the theory of lower bounds for linear forms in the logarithm of algebraic numbers with reduction methods from Diophantine approximation due to Dujella and Pethö
Summary
Before starting with the main problem of this paper, we recall some nomenclature and symbols for the convenience of the reader: The Fibonacci sequence ( Fn )n is defined by the recurrence. The Tribonacci numbers t n ( Tn )n are defined by the third-order recurrence. With initial values T0 = 0 and T1 = T2 = 1. A repdigit (short for “repeated digit”) is a number of the form a 10` − 1 !
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