Coincidences in the Padovan and Tribonacci sequences.

• Santos Hernández Hernández Unidad Académica de Matemáticas, Universidad Autónoma de Zacatecas, Campus II

Keywords: Padovan, Tribonacci sequences, Linear forms in logarithms, reduction method

Abstract

Let $(P_{n})_{n\geqslant 0}$ be the {\em Padovan sequence} given by $P_{0}=0$, $P_{1}=P_{2}=1$ and the recurrence formula $P_{n+3}=P_{n+1}+P_{n}$ for all $n\geqslant 0$. Let $(T_{n})_{n\geqslant 0}$ be the {\em Tribonacci sequence} given by $T_{0}=0$, $T_{1}=T_{2}=1$ and the recurrence formula $T_{n+3}=T_{n+2}+T_{n+1}+T_{n}$ for all $n\geqslant 0$. In this note we solve the Diophantine equation
$$P_{n}=T_{m}$$
in non-negative integers $n$, $m$. In particular, we find all the elements in the intersection of the Padovan and Tribonacci sequences.

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