Repdigits as Product of Fibonacci and Tribonacci Numbers

: 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.


Introduction
Before starting with the main problem of this paper, we recall some nomenclature and symbols for the convenience of the reader: The Fibonacci sequence (F n ) n is defined by the recurrence with initial values F 0 = 0 and F 1 = 1. The Tribonacci numbers t n (T n ) n are defined by the third-order recurrence T n+1 = T n + T n−1 + T n−2 , with initial values T 0 = 0 and T 1 = T 2 = 1. A repdigit (short for "repeated digit") is also a number of the form where ≥ 1 and a ∈ [1,9] (here, for integers a < b, we set [a, b] = {a, a + 1, . . . , b}), that is, a number with only one distinct digit in its decimal expansion.
The main subject of this work is the Diophantine equations. It is almost unnecessary to stress that these objects play an important role in the Number Theory-for example, the equations x 2 + y 2 = z 2 (Pythagoras equation), x 2 − Dy 2 = c (Pell equation), and x n + y n = z n (Fermat equation) intrigued several mathematicians at different times. It is also important to notice that their studies contributed strongly to the advance of mathematics. There are many articles that address Diophantine equations concerning Fibonacci and Lucas numbers (see, e.g., [1][2][3][4][5][6][7][8]). The linear forms in logarithms, which were probably firstly used for solving Diophantine equations in Dujella and Jadrijević [9], have proved to be a very effective tool for finding solutions to all these equations.
We point out that Luca [19] and Marques [20] proved that the largest repdigits in the Fibonacci and Tribonacci sequence are F 10 = 55 and T 8 = 44, respectively.
The aim of this paper is to continue the study of Diophantine problems involving recurrence sequences and repdigits. More precisely, we search for repdigits which are the product of Fibonacci and Tribonacci numbers with the same index. Our main result is the following: Theorem 1. The only solutions of the Diophantine equation in positive integers n, a and , with a ∈ [1,9], are (n, , a) ∈ {(1, 1, 1), (2, 1, 1), (3, 1, 4)}.
In the previous statement, the logarithmic height of a t-degree algebraic number α is defined by where a is the leading coefficient of the minimal polynomial of α (over Z), and (α (j) ) 1≤j≤t are the algebraic conjugates of α. The next lemma provides some useful properties of this function (we refer to [24] for the proof of the following facts): h(α r ) = |r| · h(α), for all r ∈ Q.
Our last tool is a reduction method provided by a variant of the well-known Baker-Davenport lemma, proved by Dujella and Pethő. For x ∈ R, set x = min{|x − n| : n ∈ Z} = |x − x | for the distance from x to the nearest integer. We refer the reader to Lemma 5 in [25] for the proof of the following lemma.
Lemma 3. For a positive integer M, let p/q be a convergent of the continued fraction of γ ∈ Q, such that q > 6 M, and let µ, A, and B be real numbers, with A > 0 and B > 1. If the number = µq − M γq is positive, then there is no solution to the Diophantine inequality in integers m, n > 0 with Now, we are ready to prove the main theorem.

Reducing the Bound
Now, we need to reduce the upper bound for n and . For that, we may suppose, with no loss of generality, that Λ > 0 (the other case is simply a mimic, considering that 0 < Λ = −Λ).
On dividing through by log(φα), we have with γ := log 10/ log(φα) and µ a := log θ a / log(φα). Clearly, γ is an irrational number (because α and φ are multiplicatively independent). Therefore, we shall denote p n /q n as the n-th convergent of the (infinite) continued fraction of γ .

Conclusions
In this paper, we solved the Diophantine equation F n T n = a(10 − 1)/9, where (F n ) n and (T n ) n are the Fibonacci and Tribonacci sequences, respectively, in positive integers n, and a, with a ∈ [1,9]. In other words, we found all repdigits (i.e., positive integers with only one distinct digit in its decimal expansion) which can be written as a product of a Fibonacci number and a Tribonacci number (both with the same index). In particular, we proved that the only repdigits with the desired property are the trivial ones, that is, those with only one digit ( = 1). To prove this result, we combined the theory of lower bounds for linear forms in the logarithm of algebraic numbers (from Baker's theory) with reduction methods from Diophantine approximation (based on the theory of continued fractions) due to Dujella