Perrin Numbers That Are Concatenations of a Perrin Number and a Padovan Number in Base b

Abstract

Let (Formula presented.) be a Padovan sequence and (Formula presented.) be a Perrin sequence. Let n, m, b, and k be non-negative integers, where (Formula presented.). In this paper, we are devoted to delving into the equations (Formula presented.) and (Formula presented.), where d is the number of digits of (Formula presented.) or (Formula presented.) in base b. We show that the sets of solutions are (Formula presented.) (Formula presented.) for the first equation and (Formula presented.) for the second equation. Our approach employs advanced techniques in Diophantine analysis, including linear forms in logarithms, continued fractions, and the properties of Padovan and Perrin sequences in base b. We investigate both the deep structural symmetries and the complex structures that connect recurrence relations and logarithmic forms within Diophantine equations involving special number sequences. © 2025 by the author