A problem on concatenated integers

Motivated by a WhattsApp message, we find out the integers \(x> y\ge 1\) such that \((x+1)/(y+1)=(x\circ(y+1))/(y\circ (x+1))\), where \(\circ\) means the concatenation of the strings of two natural numbers (for instance \(783\circ 56=78356\)). The discussion involves the equation \(x(x+1)=10y(y+...

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Bibliographic Details
Published in:arXiv.org 2021-03
Main Authors: Brunat, Josep M, Joan-Carles Lario
Format: Article
Language:English
Subjects:
Integers
Mathematical analysis
Number theory
Online Access:Get full text
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Summary:Motivated by a WhattsApp message, we find out the integers \(x> y\ge 1\) such that \((x+1)/(y+1)=(x\circ(y+1))/(y\circ (x+1))\), where \(\circ\) means the concatenation of the strings of two natural numbers (for instance \(783\circ 56=78356\)). The discussion involves the equation \(x(x+1)=10y(y+1)\), a slight variation of Pell's equation related to the arithmetic of the Dedekind ring \(\mathbb{Z}[\sqrt{10}]\). We obtain the infinite sequence \(\mathcal{S}=\{(x_n,y_n)\}_{n\ge 1}\) of all the solutions of the equation \(x(x+1)=10y(y+1)\), which tourn out to have limit \(1/\sqrt{10}\). The solutions of the initial problem on concatenated integers form the infinite subsequence of \(\mathcal{S}\) formed by the pairs \((x_n,y_n)\) such that \(x_n\) has one more digit that \(y_n\).
ISSN:2331-8422