Browkin's discriminator conjecture

TL;DR

This paper investigates Browkin's discriminator conjecture related to a specific integer sequence, proving its validity for most cases and identifying exceptions, while also extending previous results to all primes greater than or equal to 7.

Contribution

The paper proves Browkin's conjecture for all but the case n=5 and determines the discriminator values when 3 is not a primitive root, extending earlier work to all primes q ≥ 7.

Findings

The conjecture holds for all n ≠ 5.

Counterexamples exist for n=5 with infinitely many q.

Explicit formulas for D_q(n) when 3 is not a primitive root.

Abstract

Let $q\ge 5$ be a prime and put $q^*=(-1)^{(q-1)/2}\cdot q$. We consider the integer sequence $u_q(1),u_q(2),\ldots,$ with $u_q(j)=(3^j-q^*(-1)^j)/4$. No term in this sequence is repeated and thus for each $n$ there is a smallest integer $m$ such that $u_q(1),\ldots,u_q(n)$ are pairwise incongruent modulo $m$. We write $D_q(n)=m$. The idea of considering the discriminator $D_q(n)$ is due to Browkin (2015) who, in case $3$ is a primitive root modulo $q,$ conjectured that the only values assumed by $D_q(n)$ are powers of $2$ and of $q$. We show that this is true for $n\neq 5$, but false for infinitely many $q$ in case $n=5$. We also determine $D_q(n)$ in case 3 is not a primitive root modulo $q$. Browkin's inspiration for his conjecture came from earlier work of Moree and Zumalac\'arregui (2016), who determined $D_5(n)$ for $n\ge 1$, thus establishing a conjecture of Salajan. For a fixed prime $q$ their approach is easily generalized, but requires some innovations in order to deal with all primes $q\ge 7$ and all $n\ge 1$. Interestingly enough, Fermat and Mirimanoff primes play a special role in this.