Date & Time: Thursday, March 27, 2014, 16:00-17:00.
Venue: Ramanujan Hall
Title: Powers in products of terms of Pell's and Pell-Lucas Sequences
Speaker: Shanta Laishram, ISI Delhi
Abstract: It is known that there are only finitely many perfect powers in non degenerate binary recurrence sequences. However explicitly finding them is an interesting and a difficult problem for a number of binary recurrence sequences. A recent breakthrough result of Bugeaud, Mignotte and Siksek states that Fibonacci sequences $(F_n)_{n\geq 0}$ given by $F_0=0, F_1=1$ and $F_{n+2}=F_n+F_{n+1}$ for $n\geq 0$ are perfect powers only for $F_0=0, F_1=1, F_2=1, F_6=8$ and $F_{12}=144$.
In this talk, we consider another well known Pell and Pell-Lucas sequences. The Pell sequence $(u_n)_{n=0}^{\infty}$ is given by the recurrence $u_n=2u_{n-1}+u_{n-2}$ with initial condition $u_0=0, u_1=1$ and its associated Pell-Lucas sequences $(v_n)_{n=0}^{\infty}$ is given by the recurrence $v_n=2v_{n-1}+v_{n-2}$ with initial condition $v_0=2, v_1=2$.
Let $n, d, k, y, m$ be positive integers with $m\geq 2$, $y\geq 2$ and $\gcd(n,d)=1$. We prove that the only solutions of the Diophantine equation $u_{n}u_{n+d}\cdots u_{n+(k-1)d}=y^{m}$ are given by $u_7=13^2$ and $u_1u_7=13^2$ and the equation $v_{n}v_{n+d}\cdots v_{n+(k-1)d}=y^{m}$ has no solution. In fact we prove a more general result. This is a joint work with Bravo, Das and Guzman.