Binary recurrences for which powers of two are discriminating moduli

Abstract

Given a sequence of distinct positive integers w0,w1,w2, . . . and any pos- itive integer n, we define the discriminator function Dw(n) to be the smallest positive integer m such that w0, . . . ,wn−1 are pairwise incongruent modulo m. In this paper, we classify all binary recurrent sequences {wn}n 0 consisting of different integer terms such that Dw(2e) = 2e for every e ≥ 1. For all of these sequences it is expected that one can actually give a fairly simple description of Dw(n) for every n ≥ 1. For two infinite families of such sequences this has been done already in 2019 by Faye, Luca and Moree, respectively Ciolan and Moree.Please read abstract in the article.