Some Diophantine Equations involving associated Pell numbers and repdigits

TL;DR

This paper investigates the connections between repdigits and associated Pell numbers, including their differences and concatenations, using advanced number theory techniques and computational tools.

Contribution

It provides new results on representing repdigits as differences of Pell numbers and characterizes associated Pell numbers formed by concatenating repdigits.

Findings

Repdigits can be expressed as differences of associated Pell numbers.

Certain associated Pell numbers are concatenations of three repdigits.

The study employs Baker's theory and computational methods for proofs.

Abstract

In this paper, we explore the relationship between repdigits and associated Pell numbers, specifically focusing on two main aspects: expressing repdigits as the difference of two associated Pell numbers, and identifying which associated Pell numbers can be represented as the difference of two repdigits. Additionally, we investigate all associated Pell numbers which are the concatenation of three repdigits. Our proof utilizes Baker's theory on linear forms in logarithms of algebraic numbers, along with the Baker-Davenport reduction technique. The computations were carried out with the help of a simple computer program in {\it Mathematica}.