Perrin numbers that are palindromic concatenations of two repdigits

TL;DR

We address the problem of identifying Perrin numbers that are palindromic concatenations of two repdigits. The approach combines the Binet-type expression $P_n=\alpha^n+\beta^n+\gamma^n$, exact lower bounds for linear forms in logarithms (Matveev), and a Baker–Davenport–style reduction to bound the triple $(n,\ell,m)$ arising from the palindromic structure. The results show that, besides an exhaustive check up to $n\le 700$, no solutions exist for $n>700$, establishing that the unique Perrin number with the desired property is $22$. This extends analogous findings for related recurrence sequences and demonstrates the efficacy of linear forms in logarithms and continued-fraction reductions in Diophantine problems on linear recurrences.

Abstract

Let $ \{P_n\}_{n\geq 0} $ be the sequence of Perrin numbers defined by $P_0=3$, $P_1=0$,$P_2=2$ and $P_{n+3}=P_{n+1}+P_{n}$ for all $n \geq 0$. In this paper, we determine all Perrin numbers that are palindromic concatenations of two repdigits.

Theorems & Definitions (13)

• Theorem 1.1
• Lemma 2.1
• proof
• Definition 2.1: Logartihmic height
• Theorem 2.1: Matveev, matl
• Lemma 2.2: Dujella & Petho, duj
• Lemma 2.3: Lemma 7 in guz
• Lemma 3.1
• proof
• Lemma 3.2