Journal of Integer Sequences, Vol. 23 (2020), Article 20.11.3

Binary Recurrences for which Powers of Two are Discriminating Moduli


Adriaan A. de Clercq
Department of Mathematics and Applied Mathematics
University of Pretoria
Private Bag X20
Hatfield 0028
South Africa

Florian Luca
School of Maths
Wits University
1 Jan Smuts Avenue
Braamfontein 2000
Johannesburg
South Africa
and
Centro de Ciencias Matemáticas
UNAM, Morelia
Mexico

Lilit Martirosyan
Department of Mathematics and Statistics
University of North Carolina, Wilmington
601 South College Road
Wilmington, NC 28403-5970
USA

Maria Matthis
Department of Mathematics
Katharineum zu Lübeck
Königsstraße 27-31
23552 Lübeck
Germany

Pieter Moree
Max-Planck-Institut für Mathematik
Vivatsgasse 7
D-53111 Bonn
Germany

Max A. Stoumen
Department of Mathematics and Statistics
University of North Carolina, Wilmington
601 South College Road
Wilmington, NC 28403-5970
USA

Melvin Weiß
Department of Mathematics
Universität Bonn
Endenicher Allee 60
53115 Bonn
Germany

Abstract:

Given a sequence w = (wn)n≥0 of distinct positive integers w0, w1, w2,... and any positive 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 w 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 one infinite family of such sequences this has already been done by Faye, Luca, and Moree, and for another by Ciolan and Moree.

Full version:  pdf,    dvi,    ps,    latex    

(Concerned with sequences A084222 A270151.)

Received March 10 2020; revised versions received September 28 2020; October 23 2020. Published in Journal of Integer Sequences, November 28 2020.

Return to Journal of Integer Sequences home page