Perrin numbers that are concatenations of two distinct repdigits

TL;DR

This paper identifies all Perrin numbers that are formed by concatenating two different repeated digit numbers, using advanced number theory techniques.

Contribution

It explicitly determines Perrin numbers that are concatenations of two distinct repdigits, applying Baker's theory and continued fractions.

Findings

Identified all Perrin numbers as concatenations of two distinct repdigits.

Used Baker's theory to solve the Diophantine equations involved.

Provided a complete classification of such Perrin numbers.

Abstract

Let $ (P_n)_{n\ge 0}$ be the sequence of Perrin numbers defined by ternary relation $ P_0=3 $, $ P_1=0 $, $ P_2=2 $, and $ P_{n+3}=P_{n+1}+P_n $ for all $ n\ge 0 $. In this paper, we use Baker's theory for nonzero linear forms in logarithms of algebraic numbers and the reduction procedure involving the theory of continued fractions, to explicitly determine all Perrin numbers that are concatenations of two distinct repeated digit numbers.