2017/07/07 by Alexandru Ciolan, Pieter Moree
Mathematics · #math.NT #msc:11B50 #msc:11A07 #msc:11B05
published as Colloq. Math.156 (2019), 25--56 · 26 pages, 8 tables. The paper builds on an earlier paper [arXiv:1504.05718] by Moree and Zumalacarregui and for the convenience of the reader we give the complete proofs and logically this results in a considerable amount of text overlap with that paper. In the commentary in each section, we elaborate on what is similar and what is new
arxiv created 2017/07/07 · arxiv updated 2020/08/27
Let q≥ 5 be a prime and put q^*=(-1)(q-1)/2⋅ q. We consider the integer sequence uq(1),uq(2),…, with uq(j)=(3j-q^*(-1)j)/4. No term in this sequence is repeated and thus for each n there is a smallest integer m such that uq(1),…,uq(n) are pairwise incongruent modulo m. We write Dq(n)=m. The idea of considering the discriminator Dq(n) is due to Browkin (2015) who, in case 3 is a primitive root modulo q, conjectured that the only values assumed by Dq(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=5. We also determine Dq(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 (2016), who determined D5(n) for n≥ 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≥ 7 and all n≥ 1. Interestingly enough, Fermat and Mirimanoff primes play a special role in this.