Common Values of Padovan and Perrin Sequences

The integer sequence defined by 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} with initial conditions P0=P1=P2=1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$P_{0}=P_{1}=P_{2}=1$$\end{document} is known as the Padovan sequence (Pn)n∈Z\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(P_{n})_{n\in \mathbb {Z}}$$\end{document}. The Perrin sequence (Rm)m∈Z\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(R_{m})_{m\in \mathbb {Z}}$$\end{document} satisfies the same recurrence equation as the Padovan sequence but with starting values R0=3\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$R_{0}=3$$\end{document}, R1=0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$R_{1}=0$$\end{document}, and R2=2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$R_{2}=2$$\end{document}. In this note, we solve the Diophantine equation Pn=±Rm\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$P_{n}=\pm R_{m}$$\end{document} with (n,m)∈Z2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(n,m)\in \mathbb {Z}^{2}$$\end{document}.


Introduction
The Padovan numbers (P n ) n≥0 are defined by the Fibonacci-like recurrence relation P n+3 = P n+1 + P n for n ≥ 0, with initial conditions P 0 = P 1 = P 2 = 1. The first of these numbers for n ≥ 0 are 1, 1, 1, 2, 2, 3,4,5,7,9,12,16,21,28,37,49,65,86, . . . The Perrin numbers (R m ) m≥0 satisfy the same recurrence equation as Padovan numbers, but with different initial values. The first Perrin numbers for m ≥ 0 are 3, 0, 2, 3, 2, 5, 5, 7, 10, 12, 17, 22, 29, 39, 51, . . . Therefore, both sequences share the same characteristic polynomial given by X 3 −X −1. Since the constant term of this polynomial is −1, these sequences can be extended to negative indices. We call these sequences n-Padovan and n-Perrin. These sequences are linear recursive with characteristic polynomial One of the basic questions in the theory of linear recurrences is the description of the common terms of recurrences. Evertse [8] and Laurent [10] proved ineffectively that only a finite number of common terms can occur. Effective version is known only for the equations u n = u m provided that the characteristic polynomial of (u n ) has at most three roots with the same absolute value and for u n = v m when both recurrences have dominating simple and real roots. The book of Shorey and Tijdeman [15] gives detailed overview on the related results.
In 2020, Bravo et al. [3,4] established all solutions of equation in integers n, m, where (T n ) n≥0 denotes the Tribonacci sequence, which is defined by the initial terms T 0 = T 1 = 0, T 1 = 1 and by the recursion T n+3 = T n+2 + T n+1 + T n for n ≥ 0. Pethő [13] generalized the results of Bravo et al. [3,4] to generalized Fibonacci numbers, showing that equation has only finitely many solutions (n, m) ∈ Z 2 for fixed k ≥ ≥ 2, where F (k) n n≥−k+2 denotes the generalized Fibonacci sequence, which is defined by the initial values F (k) n = 0 for n = 0, . . . , −k + 2, F (k) 1 = 1 and by the k ≥ 2 fixed order recursion F for n ≥ −k + 2. Of course, for k = 3, we get the Tribonacci sequence. Unfortunately, the proof of Pethő [13] is ineffective, since it is based on the theory of S-unit equations. See also Pethő and Szalay [14].
The results of Bravo et al. [3,4] were also generalized by Pethő [12] by solving effectively the equation in positive integers n, m, where (u n ) and (v m ) denote linear recursive sequences, such that the first has dominating real root α, while the second has a dominating pair β, β of conjugate complex numbers, such that α and |β| are multiplicatively dependent.
Here, we consider solving equation for n, m ≥ 0. It has exactly 10 solutions, namely To solve completely Eq.

Results
Theorem 2.1. The only solutions of equation for n, m ≥ 0 are the 24 that we list below P −20 = 7 = R 7 ; P −25 = 10 = R 8 .

Theorem 2.2. The only solutions of equation
for n, m ≥ 0 are the 30 ones listed below

The Padovan and Perrin Sequences
We begin by recalling some properties of these ternary recurrence sequences. First, their characteristic polynomial is given by Ψ(X) = X 3 −X−1. Denoting its roots by ρ, β, γ, being ρ the only real root, an analytic expression of the kth term of the Padovan and Perrin sequences can be given, respectively, by and where , and c γ = c β .

Common Values of Padovan and Perrin Sequences
In addition, it can be shown by induction that and The following results similar to (3.4) and (3.5) for |P −n | and |R −m | were proved by using linear forms in logarithms by Bravo From the result of de Weger [  We end this section of preliminaries on the Padovan and Perrin sequences by mentioning that we can identify the automorphisms of the Galois group of the splitting field K = Q(α, β) of Ψ over Q with the permutations of the roots of Ψ, since For example, the permutation (ρβγ) corresponds to the automorphism σ ρβγ :

Linear Forms in Logarithms
For an algebraic number α of degree d over Q with minimal primitive polynomial we put for the logarithmic height of α := α (1) . The following are some basic properties of this height that will be used later without reference: Next, we give the general lower bound for linear forms in logarithms due to Matveev [11]. Let K be a number field of degree D over Q, let α 1 , . . . , α t be non-zero elements of K, and let b 1 , . . . , b t be integers. Set With this notation, the main result of Matveev [11] implies the following estimate.

Reduction Tools
Next, we remind the Baker-Davenport reduction method from Bravo, Gómez, and Luca [5, Lemma 1], which is an immediate variation of a result due to Dujella and Pethő [7, Lemma 5(a)], which turns out to be useful to reduce the bounds arising from applying Theorem 4.1.
Taking logarithms in the inequality that results above, we obtain n − 2m < 1.88 × 10 16 log n.
On the other hand, it also follows from (3.5) and (3.6) in equation (2.1) that: Taking logarithms in the resulting inequality above, we get We recorded what we just showed.
We also put M = 1.8 × 10 37 , which is an upper bound for n by Lemma 6.2. It then follows from Lemma 5.1 applied to inequality (6.10) that: where q 73 = 209509831018529557470433975207606463797 is the denominator of the first convergent of the continued fraction of κ, such that q 73 > 6M and > 0.159775. Thus, n−2m ≤ 647. If we repeat the argument after Lemma 6.1 until the upper bound for n with this new bound for n−2m, we have to replace only A 1 by 820, and we get n < 9.31 × 10 19 log n which gives n < 4.64 × 10 21 . Now, we apply Lemma 5.1 to inequality (6.10) with M = 4.64 × 10 21 . In this case with q 41 = 56787231118705906647120, we obtain that q 41 > 6 M , > 0.404707 and then n − 2m ≤ 386. From the above and Lemma 6.1, we get that n − 2 m ∈ [1,386].

The Proof of Theorem 2.2
This proof follows to a large extent the line of argument set out in the previous proof of Theorem 2.1. We omit some details. Using (3.4) and (3.7) in Eq. (2.2), we establish the following result.