Geodesic
Padovan and Perrin numbers as products of two generalized Lucas numbers
Archivum mathematicum, Tome 59 (2023) no. 4, pp. 315-337 Cet article a éte moissonné depuis la source Czech Digital Mathematics Library

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Let $P_m$ and $E_m$ be the $m$-th Padovan and Perrin numbers respectively. Let $r, s$ be non-zero integers with $r\ge 1$ and $s\in \lbrace -1, 1\rbrace $, let $\lbrace U_n\rbrace _{n\ge 0}$ be the generalized Lucas sequence given by $U_{n+2}=rU_{n+1} + sU_n$, with $U_0=0$ and $U_1=1.$ In this paper, we give effective bounds for the solutions of the following Diophantine equations \[ P_m=U_nU_k\quad \text{and}\quad E_m=U_nU_k\,, \] where $m$, $ n$ and $k$ are non-negative integers. Then, we explicitly solve the above Diophantine equations for the Fibonacci, Pell and balancing sequences.
Let $P_m$ and $E_m$ be the $m$-th Padovan and Perrin numbers respectively. Let $r, s$ be non-zero integers with $r\ge 1$ and $s\in \lbrace -1, 1\rbrace $, let $\lbrace U_n\rbrace _{n\ge 0}$ be the generalized Lucas sequence given by $U_{n+2}=rU_{n+1} + sU_n$, with $U_0=0$ and $U_1=1.$ In this paper, we give effective bounds for the solutions of the following Diophantine equations \[ P_m=U_nU_k\quad \text{and}\quad E_m=U_nU_k\,, \] where $m$, $ n$ and $k$ are non-negative integers. Then, we explicitly solve the above Diophantine equations for the Fibonacci, Pell and balancing sequences.
DOI : 10.5817/AM2023-4-315
Classification : 11B39, 11J86
Keywords: generalized Lucas numbers; linear forms in logarithms; reduction method 
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Adédji, Kouèssi Norbert; Odjoumani, Japhet; Togbé, Alain. Padovan and Perrin numbers as products of two generalized Lucas numbers. Archivum mathematicum, Tome 59 (2023) no. 4, pp. 315-337. doi: 10.5817/AM2023-4-315

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