Perrin numbers that are concatenations of two repdigits

Let (Pn)n≥0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ (P_n)_{n\ge 0}$$\end{document} be the sequence of Perrin numbers defined by ternary relation P0=3\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ P_0=3 $$\end{document}, P1=0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ P_1=0 $$\end{document}, P2=2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ P_2=2 $$\end{document}, and Pn+3=Pn+1+Pn\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ P_{n+3}=P_{n+1}+P_n $$\end{document} for all n≥0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ n\ge 0 $$\end{document}. In this paper, we use Baker’s theory for nonzero linear forms in logarithms of algebraic numbers and the reduction procedure involving the theory of continued fractions, to explicitly determine all Perrin numbers that are concatenations of two repeated digit numbers.

The Padovan numbers and Perrin numbers share many similar properties. In particular, they have the same recurrence relation, the difference being that the Padovan numbers are initialized via Pad(0) = 0 and Pad(1) = Pad(2) = 1. This means that the two sequences also have the same characteristic equation.
Despite the similarities, the two sequences also have some stark differences. For instance, the Perrin numbers satisfy the remarkable divisibility property that if n is prime, then n divides P n . One can easily confirm that this does not hold for the Padovan numbers.
Inspired by the second author's result in [5], we study and completely solve the Diophantine equation

Preliminary results
In this section, we collect some facts about Perrin numbers and other preliminary lemmas that are crucial to our main argument.

Some properties of the Perrin numbers
Recall that the characteristic equation of the Perrin sequence is given by φ(x) := x 3 − x − 1 = 0, with zeros α, β and γ = β given by where For all n ≥ 0, Binet's formula for the Perrin sequence tells us that the nth Perrin number is given by Numerically, the following estimates hold for the quantities {α, β, γ }: It follows that the complex conjugate roots β and γ only have a minor contribution to the right-hand side of Eq. (2.1). More specifically, let e(n) := P n − α n = β n + γ n . Then, |e(n)| < 3 α n/2 for all n ≥ 1.
The following estimate also holds: Lemma 2.1 Let n ≥ 2 be a positive integer. Then α n−2 ≤ P n ≤ α n+1 . Lemma 2.1 follows from a simple inductive argument, and the fact that α 3 = α + 1, from the characteristic polynomial φ.
Let K := Q(α, β) be the splitting field of the polynomial φ over Q. Then, [K : Q] = 6 and [Q(α) : Q] = 3. We note that the Galois group of K/Q is given by We therefore identify the automorphisms of G with the permutation group of the zeroes of φ. We highlight the permutation (αβ), corresponding to the automorphism σ : α → β, β → α, γ → γ , which we use later to obtain a contradiction on the size of the absolute value of a certain bound.

Linear forms in logarithms
Our approach follows the standard procedure of obtaining bounds for certain linear forms in (nonzero) logarithms. The upper bounds are obtained via a manipulation of the associated Binet's formula for the given sequence. For the lower bounds, we need the celebrated Baker's theorem on linear forms in logarithms. Before stating the result, we need the definition of the (logarithmic) Weil height of an algebraic number.
Let η be an algebraic number of degree d with minimal polynomial where the leading coefficient a 0 is positive and the η j 's are the conjugates of η. The logarithmic height of η is given by Note that, if η = p q ∈ Q is a reduced rational number with q > 0, then the above definition reduces to h(η) = log max{| p|, q}.
We list some well-known properties of the height function below, which we shall subsequently use without reference We quote the version of Baker's theorem proved by Bugeaud et al. [1,Theorem 9.4].

Reduction procedure
The bounds on the variables obtained via Baker's theorem are usually too large for any computational purposes. To get further refinements, we use the Baker-Davenport reduction procedure. The variant we apply here is the one due to Dujella and Pethő [6, Lemma 5a]. For a real number r , we denote by r the quantity min{|r − n| : n ∈ Z}, the distance from r to the nearest integer.

Lemma 2.3 [6]
Let κ = 0, A, B and μ be real numbers, such that A > 0 and B > 1. Let M > 1 be a positive integer and suppose that p q is a convergent of the continued fraction expansion of κ with q > 6M. Let If ε > 0, then there is no solution of the inequality Lemma 2.3 cannot be applied when μ = 0 (since then ε < 0). In this case, we use the following criterion due to Legendre, a well-known result from the theory of Diophantine approximation. For further details, we refer the reader to the books of Cohen [2,3].

Lemma 2.4 [2,3] Let κ be real number and x, y integers, such that
Then, x/y = p k /q k is a convergent of κ. Furthermore, let M and N be a non-negative integers, such that q N > M. Then, putting a(M) := max{a i : i = 0, 1, 2, . . . , N }, the inequality holds for all pairs (x, y) of positive integers with 0 < y < M.
We will also need the following lemma by Gúzman Sánchez and Luca [8,Lemma 7]:

The initial bound on n
We note that Eq. (1.1) can be rewritten as The next lemma relates the sizes of n and + m.
Taking the logarithm on both sides, we get n log α < ( + m) log 10 + 2 log α.
We proceed to examine (3.1) in two different steps as follows.
Step 1. From Eqs. (2.1) and (3.1), we have Thus, we have where we used the fact that n > 500. Dividing both sides by d 1 × 10 +m , we get We let We shall proceed to compare this upper bound on | 1 | with the lower bound we deduce from Theorem 2.2. Note that 1 = 0, since this would imply that α n = 10 +m ×d 1 9 . If this is the case, then applying the automorphism σ on both sides of the preceding equation and taking absolute values, we have that σ 10 +m × d 1 9 = |σ (α n )| = |β n | < 1, which is false. We thus have 1 = 0. With a view towards applying Theorem 2.2, we define the following parameters: Note that, by Lemma 3.1, we have that + m < n. Thus, we take B = n. We note that K := Q(η 1 , η 2 , η 3 ) = Q(α). Hence, D := [K : Q] = 3. We note that h(η 1 ) = h 9 d 1 ≤ 2h(9) = 2 log 9 < 5.
After a simplification, we obtain the bound n < 4.2 × 10 31 .
The following lemma summarizes what we have proved so far:

The reduction procedure
We note that the bounds from Lemma 3.2 are too large for computational purposes. However, with the help of Lemma 2.3, they can be considerably sharpened. The rest of this section is dedicated towards this goal. We proceed as in [5]. Using Eq. (3.3), we define the quantity 1 as Equation (3.2) can thus be rewritten as If ≥ 2, then the above inequality is bounded above by 1 2 . Recall that if x and y are real numbers, such that |e x − 1| < y, then x < 2y. We therefore conclude that | 1 | < 92 10 . Equivalently Dividing throughout by log α, we get − n log α + m log 10.
Therefore, we have that n ≤ 294. This contradicts our assumption that n > 500. Hence, Theorem 1.1 is proved.
Funding The authors acknowledge the financial support for publication preparations costs extended from NORHED-II project "Mathematics for Sustainable Development (MATH4SD), 2021-2026" at Makerere University in collaboration with the University of Dar es Salaam and University of Bergen in Norway.

Conflict of interest
The authors have not disclosed any competing interests.

Acknowledgements
The authors thank the referee and the editor for the useful comments and suggestions that have greatly improved on the quality of presentation of the current paper.
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