On Concatenations of Two Padovan and Perrin Numbers

Let (Pk)k≥0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(P_{k})_{k\ge 0}$$\end{document} and (Rk)k≥0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(R_{k})_{k\ge 0}$$\end{document} be the Padovan and Perrin sequences. In this paper, we found that all Padovan numbers, which are concatenations of two Padovan numbers are 12, 21, 37, 49, 265, 465. Moreover, we showed that the only Perrin number, which is concatenations of two Perrin numbers is 22. That is, we solved the Diophantine equations Pk=10dPm+Pn\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ P_{k}=10^{d}P_{m}+P_{n}$$\end{document} and Rk=10dRm+Rn\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$R_{k}=10^{d}R_{m}+R_{n}$$\end{document} in positive integers (k, m, n),  where d denotes the number of digits of Pn\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$P_{n}$$\end{document} and Rn\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$R_{n}$$\end{document}, respectively. The proofs based on Baker’s theory and we used linear forms in logarithms and reduction method to solve of these Diophantine equations.