BROWKIN'S DISCRIMINATOR CONJECTURE

Let q >= 5 be a prime and put q* = (-1) ((q-1 )/2) (.) q. We consider the integer sequence u(q)(1), u(q) (2), ... 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), ...,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 5, but false for infinitely many q in case n not equal 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 Zumalacarregui, who determined D-5(n) for n >= 1, thus proving 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 >= 7 and all n >= 1. Interestingly enough, Fermat and Mirimanoff primes play a special role in this.