Padovan numbers which are palindromic concatenations of two distinct repdigits

In this paper we determine all Padovan numbers that are palindromic concatenations of two distinct repdigits.


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A repdigit (in base 10) is a positive integer N that has only one distinct digit. That is, the decimal expansion of N takes the form for some positive integers d and with 0 ≤ d ≤ 9 and ≥ 1. This paper is a contribution to the rather well studied topic of Diophantine properties of certain linear recurrence sequences. More specifically, our paper is a variation on the theme focusing on representations of terms of a recurrent sequence as concatenations of members of another (possibly the same) sequence. For a general study of the results underpinning this topic, we direct the reader to the paper [2] by Luca and Banks, wherein (as a consequence of their level of generality) some ineffective (but finiteness) results were obtained on the number of terms of certain binary recurrent sequences whose digital representation consists of members of the same sequence. In Ref. [1], the authors considered Fibonnaci numbers which are concatenations of two repdigits (in base 10) and showed that the largest such number is F 14 = 377. Recently, diophantine equations involving Padovan numbers and repdigits have also been studied. In Ref. [12], the authors found all repdigits that can be written as a sum of two Padovan numbers. This result was later extended to repdigits that are a sum of three Padovan numbers by the second author in Ref. [8]. In Ref. [9], in the direction similar to the one in Ref. [ Other related interesting results in this research direction include: the result of Bednařík and Trojovská [3], the result of Boussayoud, et al. [4], the result of Bravo and Luca [5], the result of the second author [7], the result of Erduvan and Keskin [11], the result of Rayaguru and Panda [16], the results of Trojovský [17,18], and the result of Qu and Zeng [15]. A natural continuation of the result in Ref. [9] would be a characterization of palindromic Padovan numbers. As a first step in this direction, we (for the time being) consider the (more restrictive) Diophantine equation Our result is the following.

Theorem 1
The only Padovan numbers which are palindromic concatenations of two distinct repdigits are P n ∈ {151, 616}.

Preliminary results
In this section we collect some facts about Padovan numbers and other preliminary lemmas that are crucial to our main argument. This preamble to the main result is similar to the one in Ref. [9] and is included here for the sake of completeness.

Some properties of the Padovan numbers
Recall that the characteristic equation of the Padovan sequence is given by φ(x) := x 3 − x − 1 = 0, with roots α, β, and γ = β given by: where For all n ≥ 0, Binet's formula for the Padovan sequence tells us that the nth Padovan number is given by where =b.
Lemma 1 follows from a simple inductive argument. 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 Therefore, we identify the automorphisms of G with the permutation group of the zeroes of φ. We shall find particular use for the permutation (αβ), corresponding to the automorphism σ : α → β, β → α, γ → γ .

Linear forms in logarithms
Like many proofs of similar results, the crucial steps in our argument involve obtaining certain bounds on linear forms in (nonzero) logarithms. The upper bounds usually follow easily from a manipulation of the associated Binet's formula for the sequence in question. 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 ([6], Theorem 9.4).

Baker-Davenport reduction
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 Pethö ( [10], Lemma 5a). For a real number r , we denote by r the quantity min{|r − n| : n ∈ Z}, which is the distance from r to the nearest integer.
Lemma 2 Let κ = 0, and A, B, μ be real numbers with 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 We will also need the following lemma by Gúzman Sánchez and Luca ([13], Lemma 7):

The low range
With the help of a simple computer program in Mathematica, we checked all the solutions to the Diophantine equation (1) in the ranges d 1 = d 2 ∈ {0, 1, 2, . . . , 9}, d 1 > 0 and 1 ≤ , m ≤ n ≤ 1000. We found only the solutions stated in Theorem 1. Here onwards, we assume that n > 1000.

The initial bound on n
We note that (1) can be rewritten as The next lemma relates the sizes of n and 2 + m.
We proceed to examine (5) in three different steps as follows.
Step 3. We rewrite Eq. (5) as Therefore, Consequently, As before, we have that 3 = 0 since we would have that Applying the automorphism σ from the Galois group G on both sides of the above equation and then taking absolute values, we have that which is false. We would now like to apply Theorem 2 to 3 . To this end, we let: As in the previous cases, we can take B := n and D := 3. We note that ≤ 5 log 9 + log 23 3 + ( + m) log 10 + m log 10 ≤ 6 log 9 + ( + m) log 10 + m log 10.
After a simplification, we obtain the (rather loose) bound

The reduction procedure
The bounds obtained in Lemma 5 are too large to be useful computationally. Thus, we need to reduce them. To do so, we apply Lemma 2 as follows. First, we return to the inequality (6) and put The inequality (6) can be rewritten as If we assume that ≥ 2, then the right-hand side of the above inequality is at most 28/100 < 1/2. The inequality |e z − 1| < x for real values of x and z implies that z < 2x. Thus, This implies that Dividing through the above inequality by log α gives From the inequality (8), we have that Assume that m ≥ 2, then the right-hand side of the above inequality is at most 19/100 < 1/2. Thus, we have that which implies that Dividing through by log α gives Thus, we apply Lemma 2 with the quantities: We take the same κ and its convergent p/q = p 141 /q 141 as before. Since + m < 2 + m, we set M := 10 66 as the upper bound on + m. With the help of a simple computer program in Mathematica, we get that ε > 0.0000918806, and therefore, m ≤ log((38/ log α)q/ε) log 10 < 73.
Again, we apply Lemma 2 with the quantities: We take the same κ and its convergent p/q = p 141 /q 141 as before. Since < 2 + m, we choose M := 10 66 as the upper bound for . With the help of a simple computer program in Mathematica, we get that ε > 0.00000594012, and thus, n ≤ log((6/ log α)q/ε) log α < 602.
Thus, we have shown that n ≤ 602, contradicting our assumption that n > 1000. Therefore, Theorem 1 holds.