How can the product of two binary recurrences be constant?

Résumé

Let $\omega$ denote an integer. This paper studies the equation $G_nH_n=\omega$ in the integer binary recurrences $\{G\}$ and $\{H\}$ satisfy the same recurrence relation. The origin of the question gives back to the more general problem $G_nH_n+c=x_{kn+l}$ where $c$ and $k\ge0,~l\ge0$ are fixed integers, and the sequence $\{x\}$ is like $\{G\}$ and $\{H\}$. The case of $k=2$ has already been solved (\cite{KLSz}) and now we concentrate on the specific case $k=0$.