Salajan's conjecture on discriminating terms in an exponential sequence

TL;DR

This paper proves a 2012 conjecture by Sabin Salajan regarding the discriminator function for a specific exponential sequence, providing a complete characterization of its behavior.

Contribution

It confirms Salajan's conjecture by characterizing the discriminator of the sequence defined by $u_j=(3^j-5(-1)^j)/4$, advancing understanding of discriminators in exponential sequences.

Findings

Discriminator $D_v(n)$ for the sequence is explicitly characterized.

The paper confirms Salajan's 2012 conjecture.

Provides a complete description of the discriminator's properties.

Abstract

Given a sequence of distinct positive integers $v_1,v_2,\ldots$ and any positive integer $n$, the discriminator $D_v(n)$ is defined as the smallest positive integer $m$ such $v_1,\ldots,v_n$ are pairwise incongruent modulo $m$. We consider the discriminator for the sequence $u_1,u_2,\ldots$, where $u_j$ equals the absolute value of $((-3)^j-5)/4$, that is $u_j=(3^j-5(-1)^j)/4$. We prove a 2012 conjecture of Sabin Salajan characterizing the discriminator of the sequence $u_1,u_2,\ldots$.