Perrin numbers that are concatenations of two distinct repdigits

Let $ (P_n)_{n\ge 0}$ be the sequence of Perrin numbers defined by ternary relation $ P_0=3 $, $ P_1=0 $, $ P_2=2 $, and $ P_{n+3}=P_{n+1}+P_n $ for all $ n\ge 0 $. 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 distinct 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 P ad(0) = 0 and P ad(1) = P ad(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: where d 1 = d 2 ∈ {0, 1, 2, . . . , 9}, d 1 > 0, ℓ, m ≥ 1, and n ≥ 0.
We ignore the d 1 = d 2 case for the time being, since it has been covered within a more general context in an upcoming paper, where we study the reverse question of repdigits which are sums of Perrin numbers. In any case, the only such Perrin number which is a solution of the above Diophantine equation is P 11 = 22.
Our main result is the following.

Preliminary Results.
In this section we collect some facts about Perrin numbers and other preliminary lemmas that are crucial to our main argument.
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}: 1.32 <α < 1.33, It follows that the complex conjugate roots β and γ only have a minor contribution to the right hand side of equation (2). 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 1. Let n ≥ 2 be a positive integer. Then Lemma 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 α j 's are the conjugates of α. 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, Mignotte and Siksek ( [1], Theorem 9.4).

Reduction procedure.
The bounds on the variables obtained via Baker's theorem are usually too large for any computational purposes. In order 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 (Dujella, Pethő, [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 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 3 (Legendre, [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 nonnegative 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 .

The low range.
We used a computer program in Mathematica to check all the solutions of the Diophantine equation (1) for the parameters d 1 = d 2 ∈ {0, . . . , 9}, d 1 > 0 and 1 ≤ ℓ, m and 1 ≤ n ≤ 500. We only found the solutions listed in Theorem 1. Henceforth, we assume n > 500.

The initial bound on n.
We note that equation (1) can be rewritten as The next lemma relates the sizes of n and ℓ + m.
We proceed to examine (3) in two different steps as follows.
Step 1. From equations (2) and (3), we have that Hence, Thus, we have that where we used the fact that n > 500. Dividing both sides by d 1 × 10 ℓ+m , we get We let We shall proceed to compare this upper bound on |Γ 1 | with the lower bound we deduce from Theorem 2. Note that Γ 1 = 0, since this would imply that α n = 10 ℓ+m ×d 1

9
. If this is the case, then applying the automorphism σ on both sides of the preceeding equation and taking absolute values, we have that 10 ℓ+m × d 1 9 = |σ(α n )| = |β n | < 1, which is false. We thus have that Γ 1 = 0.
In order to determine what A 1 will be, we need to find the find the maximum of the quantities h(η 1 ) and | log η 1 |.
With the notation of Lemma 4, we let r := 2, L := n and H := 1.10 × 10 44 and notice that this data meets the conditions of the lemma. Applying the lemma, we have that After a simplification, we obtain the bound n < 4.6 × 10 48 .

The reduction procedure.
We note that the bounds from Lemma 6 are too large for computational purposes. However, with the help of Lemma 2, they can be considerably sharpened. The rest of this section is dedicated towards this goal. We proceed as in [5].
Thus, n ≤ 454 in both cases. This contracts our assumption that n > 500. Hence, Theorem 1 is proved.