Multiplicative Dependent Pairs in the Sequence of Padovan Numbers

The Padovan sequence {P-n}(n >= 0) is a ternary recurrent sequence defined recursively by the relation P-n = Pn-2 + Pn-3 with initials P-0 = P-1 = P-2 = 1. In this note, we search all pairs of multiplicative dependent vectors whose coordinates are Padovan numbers. For this purpose, we apply Matveev's theorem to find the lower bounds of the non-zero linear forms in logarithms. Techniques involving the LLL algorithm and the theory of continued fraction are utilized to reduce the bounds.