  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.
