Discriminators of Integer Sequences

Abstract

The discriminator of an integer sequence \textbf{s} = $(s(n))_{n \geq 0}$, first introduced by Arnold, Benkoski and McCabe in 1985, is the function $D_s (n)$ that maps the integer $n \geq 1$ to the smallest positive integer $m$ such that the first $n$ terms of \textbf{s} are pairwise incongruent modulo $m$. In this thesis, we provide a basic overview of discriminators, examining the background literature on the topic and presenting some general properties of discriminators.

We also venture into various computational aspects relating to discriminators, such as providing algorithms to compute the discriminator, and establishing an upper bound on the discriminator growth rate. We provide a complete characterization of sequences whose discriminators are themselves, and also explore the problem of determining whether a given sequence is a discriminator of some other sequence with some partial results and algorithms.

We briefly discuss some $k$-regular sequences, characterizing the discriminators for the evil and odious numbers, and show that $k$-regular sequences do not necessarily have $k$-regular discriminators. We introduce the concept of shift-invariant discriminators, i.e. discriminators that remain the same even if the original sequence is shifted, and present a class of exponential sequences with this property. Finally, we provide a complete characterization of quadratic sequences with discriminator $p^{\lceil \log_p n \rceil}$ for primes $p \neq 3$, and provide some partial results for the case of $p = 3$.