Members of Narayana's cow sequence that are concatenations of two repdigits

TL;DR

This paper characterizes all Narayana's cow sequence numbers that are formed by concatenating two repdigits, employing advanced Diophantine approximation techniques to solve the problem.

Contribution

It provides a complete classification of Narayana numbers that are concatenations of two repdigits, using methods from transcendence theory and Diophantine approximation.

Findings

Identifies all Narayana numbers that are concatenations of two repdigits.

Uses lower bounds for linear forms in logarithms in the proof.

Applies Baker-Davenport reduction method to refine results.

Abstract

Let $ (N_n)_{n\ge 0}$ be the Narayana's cow sequence defined by a third-order recurrence relation $ N_0=0,\ N_1= N_2=1 $, and $ N_{n+3}=N_{n+2}+N_n $ for all $ n\ge 0 $. In this paper, we determine all Narayana numbers that are concatenations of two repdigits. The proof of our main theorem uses lower bounds for linear forms in logarithms and a version of the Baker-Davenport reduction method in Diophantine approximation.