Padovan numbers that are concatenations of two repdigits

Let $ (P_{n})_{n\ge 0} $ be the sequence of Padovan numbers defined by $ P_0=0 $, $ P_1 =1=P_2$, and $ P_{n+3}= P_{n+1} +P_n$ for all $ n\ge 0 $. In this paper, we find all Padovan numbers that are concatenations of two repdigits.

A repdigit is a positive integer N that has only one distinct digit when written in its decimal expansion. That is, N is of the form for some positive integers d, ℓ with 0 ≤ d ≤ 9 and ℓ ≥ 1. The sequence of repdigits is sequence A010785 on the OEIS. Diophantine equations involving repdigits and Padovan numbers have been considered in various papers in the recent years. For example: in [5], García Lomelí and Hernández Hernández found all repdigits that can be written as a sum of two Padovan numbers; in [3], the author found all repdigits that can be written as a sum of three Padovan numbers.

Main Result
In this paper, we study the problem of finding all Padovan numbers that are concatenations of two repdigits. More precisely, we completely solve the Diophantine equation Our main result is the following. This paper is inspired by the result of Alahmadi, Altassan, Luca, and Shoaib [1], in which they find all Fibonacci numbers that are concatenations of two repdigits. Our method of proof involves the application of Baker's theory for linear forms in logarithms of algebraic numbers, and the Baker-Davenport reduction procedure. Computations are done with the help of a computer program in Mathematica.

Linear forms in logarithms.
Let η be an algebraic number of degree d with minimal primitive polynomial over the integers where the leading coefficient a 0 is positive and the η (i) 's are the conjugates of η. Then the logarithmic height of η is given by In particular, if η = p/q is a rational number with gcd(p, q) = 1 and q > 0, then h(η) = log max{|p|, q}. The following are some of the properties of the logarithmic height function h(·), which will be used in the next section of this paper without reference: We recall the result of Bugeaud, Mignotte, and Siksek ([2], Theorem 9.4, pp. 989), which is a modified version of the result of Matveev [7], which is one of our main tools in this paper.
Theorem 3.1. Let η 1 , . . . , η t be positive real algebraic numbers in a real algebraic number field K ⊂ R of degree D, b 1 , . . . , b t be nonzero integers, and assume that 3.3. Reduction procedure. During the calculations, we get upper bounds on our variables which are too large, thus we need to reduce them. To do so, we use some result from the theory of continued fractions. For a nonhomogeneous linear form in two integer variables, we use a slight variation of a result due to Dujella and Pethő ([4], Lemma 5a). For a real number X, we write X := min{|X − n| : n ∈ Z} for the distance from X to the nearest integer.
Lemma 3.1. Let M be a positive integer, p q be a convergent of the continued fraction expansion of the irrational number τ such that q > 6M , and A, B, µ be some real numbers with A > 0 and B > 1. Furthermore, let ε := µq − M τ q . If ε > 0, then there is no solution to the inequality in positive integers u, v, and w with u ≤ M and w ≥ log(Aq/ε) log B .
We have just proved the following lemma.

4.3.
Reducing the bounds. The bounds given in Lemma 4.2 are to large to carry out meaningful compution. Thus, we need to reduce them. To do so, we apply Lemma 3.1 as follows. First, we return to (4.4) and put The inequality (4.4) can be rewritten as e −Γ1 − 1 < 30 10 ℓ1 . Assume that ℓ 1 ≥ 2, then the right-hand side in the above inequality is at most 3/10 < 1/2. The inequality |e x − 1| < y for real values of x and y implies that x < 2y. Thus, |Γ 1 | < 60 10 ℓ1 , which implies that (ℓ 1 + ℓ 2 ) log 10 − n log α − log 9a d 1 < 60 10 ℓ1 .
Thus, we have that n ≤ 446, contradicting the working assumption that n > 500. Hence, Theorem 2.1 is proved.