No label defined (Q6645086)

\noindent Let \( (U_n)_{n\ge 1} \) be a linearly recurrent sequence of integers that satisfies a recurrence relation of the form\N\begin{align*}\NU_{n+k}=aU_{n+k-1}+\cdots+a_kU_n \quad \text{for all}~~n\ge 1,\N\end{align*}\Nwith integers \( a_1, \ldots, a_k \), where \( U_1, \ldots, U_k \) are integers. Let \( \phi(n) \) and \( \sigma(m) \) be the Euler function and sum of divisors function of the positive integer \( m \), respectively. In the paper under review, the authors prove the following theorem, which is the main result in the paper.\N\NTheorem 1. Let \( (U_n)_{n\ge 1} \) be a non-degenerate linearly recurrent sequence of integers such that \( |U_n| \) is not a polynomial in \( n \) for all large \( n \) and let \( x \) be a large real number. Then, the inequality\N\begin{align*}\N\phi(|U_n|)\ge |U_{\phi(n)}|,\N\end{align*}\Nfails on a set of positive integers \( n\le x \) of cardinality \( O_U\left(\dfrac{x}{\log x}\right) \). A similar statement holds for the positive integers \( n\le x \) for which the inequality\N\begin{align*}\N\sigma(|U_n|)\le |U_{\sigma(n)}|,\N\end{align*}\Nfails. \N\NThe proof of Theorem 1 heavily employees a combination of techniques in number theory, the Subspace theorem and results on linearly recurrent sequences.