On the Euler function of \(Y\)-coordinates of Pell equations and repdigits
type: Item
instance of: scholarly article
title: On the Euler function of \(Y\)-coordinates of Pell equations and repdigits
zbMATH Open document ID: 1485.11073
author: Florian Luca
author: Adel Alahmadi
published in: Results in Mathematics
publication date: 2022-02-03T00:00:00Z
review text: Let \(d\geq 1\) be an integer, \(c\in\{ 1,4\}\) and \(\varphi(x)\) be the Euler function. Let \(\{ Y_m\}_{m\geq 1}\) be the sequence of the \(Y\)-coordinates of the Pell equation \(X^2-dY^2=\pm c\). Assume further \(X_1^2-dY_1^2=- c\). Then the authors prove that all solutions of the equation \(\varphi(Y_m)=\frac 19D(10^n-1)\) where \(D\in\{ 1,\ldots ,9\}\) satisfy one of the following conditions 1. either \(m\leq 100\) or, 2. \(m=p\) is an odd prime, \(Y_1\in\{ 1,2\}\) and \(Y_p/Y_1=q\) is also an odd prime. Further, \(D=8\) and \(n\equiv 2 \pmod 3\).
Mathematics Subject Classification ID: 11D61
Mathematics Subject Classification ID: 11A25
Mathematics Subject Classification ID: 11B39
Mathematics Subject Classification ID: 11A63
zbMATH DE Number: 7468844
zbMATH Keywords: Pell equation
zbMATH Keywords: Euler function
reviewed by: Jaroslav Hančl
MaRDI profile type: Publication
full work available at URL: s00025-021-01594-0
OpenAlex ID: W4205205209
cites work: Q5435625
cites work: Q5230580
DOI: 10.1007/S00025-021-01594-0
description: scientific article; zbMATH DE number 7468844