Seminar at SMU Delhi

August 26, 2015 (Wednesday) , 3:30 PM at Webinar
Speaker: Pranabesh Das, Indian Statistical Institute, Delhi
Title: On some equations involving binary recurrence sequences
Abstract of Talk

In mathematics, mostly in number theory and combinatorics, binary recurrence sequences is an important topic which has been well studied. We define binary recurrence sequence by $U_n=AU_{(n-1)}+BU_{(n-2)}$ with $A,B$ coprime integers and with some initial values of $U_0$ and $U_1$. Fibonacci sequence is a well known example of such a sequence.

In this talk we consider two problems. In the first part we will consider another example of binary recurrence sequence namely Pell and Pell-Lucas sequences. The Pell sequence $U_n$ is given by the recurrence ${U_n}= 2{U_{(n-1)}}+ {U_{(n-2)}}$ with initial condition $U_0 = 0$, $U_1 = 1$ and its associated Pell-Lucas sequences $V_n$ is given by the recurrence ${V_n}= 2{V_{(n-1)}}+ {V_{(n-2)}}$ with initial condition $V_0 = 2$, $V_1 = 2$.

Let $n, d, k, y, m$ be positive integers with $m\geq2$, $y\geq2$ and $\gcd(n, d) = 1$. We prove that the only solutions of the Diophantine equation $${U_{n}U_{(n+d)}...U_{(n-d+kd)}} = {y^{m}}$$ are given by $U_7 = 13^2$ and $U_1 U_7 = 13^2$ and the equation $${V_{n}V_{(n+d)}...V_{(n-d+kd)}} =y^m$$ has no solution. In fact we prove more general results.This is a joint work with S. Laishram, S. Guzman, J. Bravo.

In the last part, we will consider some equations involving general Lucas sequences $(U_n)$ and Associated Lucas sequences $(V_n)$. In particular, we will solve the following equations $${\binom mk} = \frac{U_{m}U_{m-1}...U_{m-k+1}}{U_1U_2...U_k}={m^r}{k^s}$$ and $${\binom mk} = \frac{V_{m}V_{m-1}...V_{m-k+1}}{V_1V_2...V_k}={m^r}{k^s}$$ in positive integer variables $m, k, r, s$ with $0\leq{k}\leq{m}$. This is a joint work with S. Laishram.