Browkin’s discriminator conjecture
Volume 156 / 2019
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 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 $n\not =5$, 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 Zumalacárregui, who determined $D_5(n)$ for $n\ge 1$, thus proving a conjecture of Sălăjan. 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.
