On the Euler function of \(Y\)-coordinates of Pell equations and repdigits
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 \(n\equiv 2 \pmod 3\).
scientific article; zbMATH DE number 7468844