On Padovan numbers that are perfect powers of Lucas numbers

Let $(P_n)$ and $(L_n)$ be the Padovan and Lucas sequences. In this paper, we found all Padovan numbers that were perfect powers of Lucas numbers using advanced mathematical tools such as linear forms in logarithms and the reduction method. The results for this equation were

$$9=P_9=L_2^2=3^2,16=P_{11}=L_3^2=4^2,49=P_{15}=L_4^2=7^2.$$

Moreover, the intersection between the Padovan and Lucas sequences revealed that

$$1=P_2=L_1,2=P_3=L_0,2=P_4=L_0,3=P_5=L_2,4=P_6=L_3,7=P_8=L_4.$$

This work not only solved a specific open problem in exponential Diophantine equations but also highlighted the wider applicability of the methods used. The results paved the way for further research on perfect powers in other linear iteration sequences and their intersections, emphasizing the importance of such research in modern mathematics.