Pell or Pell-Lucas numbers as concatenations of two repdigits in base $b$

TL;DR

This paper characterizes all Pell and Pell-Lucas numbers that can be expressed as concatenations of two repdigits in bases 2 through 10, identifying the largest such numbers in each sequence.

Contribution

It provides a complete classification of Pell and Pell-Lucas numbers as concatenations of two repdigits in specified bases, including the largest examples.

Findings

Largest Pell number as concatenation: P_{11} = 5741

Largest Pell-Lucas number as concatenation: Q_5 = 82

Complete characterization for bases 2 to 10

Abstract

Let $b$ be a positive integer such that $2 \leq b \leq 10$. In this study, we find all Pell or Pell-Lucas numbers as concatenations of two repdigits in base $b$. As a corollary, it is show that the largest Pell or Pell-Lucas numbers which can be expressible as a concatenations of two repdigits in base $b$ are $P_{11} = 5741$ and $Q_5 = 82$, respectively.