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 = P n –2 + P n –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.