A problem on concatenated integers

Abstract

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 $y_n$.