Binary recurrences for which powers of two are discriminating moduli

Abstract

Given a sequence of distinct positive integers $w_0,w_1,w_2,\dots$ and any positive integer $n$, we define the discriminator function ${\mathcal{D}}_{\mathbf{w}}(n)$ to be the smallest positive integer $m$ such that $w_0,\dots ,w_{n-1}$ are pairwise incongruent modulo $m$. In this paper, we classify all binary recurrent sequences $\{w_n{\}}_{n\ge 0}$ consisting of different integer terms such that ${\mathcal{D}}_{\mathbf{w}}(2^e)=2^e$ for every $e\ge 1.$ For all of these sequences it is expected that one can actually give a fairly simple description of ${\mathcal{D}}_{\mathbf{w}}(n)$ for every $n\ge 1.$ For two infinite families of such sequences this has been done already in 2019 by Faye, Luca and Moree, respectively Ciolan and Moree.

10 pages, 2 tables, final version