Description
Online Number Theory Seminar
Abstract: Let $d>1$ be an integer which not a square and $(X_n,Y_n)$ be the $n$th solution of the Pell equation $X^2-dY^2=\pm 1$. Given an interesting set of positive integers $U$, we ask how many positive integer solutions $n$ can the equation $Y_n\in U$ have. Under mild assumptions on $U$ (for example, when $1\in U$ and $U$ contains infinitely many even integers), the equation $Y_n\in U$ has two solutions $n$ for infinitely many $d$. We show that this is best possible whenever $U$ is the set of values of a binary recurrent sequence $\{u_m\}_{m\ge 1}$ with real roots and $d$ is large enough (with respect to $U$). We also treat the cases when $U$ is one of the sets $\{2^n-1: n\ge 1\}$, $\{F_n: n\ge 1\}$ and $\{L_n: n\ge 1\}$, where $F_n$ and $L_n$ are the $n$th Fibonacci and Lucas numbers. For example, $Y_n=2^m-1$ has at most two positive integer solutions $(n,m)$ for all $d$ and each of the equations $Y_n=F_m$ or $Y_n=L_m$ has exactly two solutions $(n,m)$ except for $d=2$, in which case it has exactly three solutions both when Fibonacci or Lucas numbers are involved. The proofs use linear forms in logarithms.
For access please contact the organizers (ntrg[at]science.unideb.hu).