Ciolan, Alexandru and Moree, Pieter (2019). BROWKIN'S DISCRIMINATOR CONJECTURE. Colloq. Math., 156 (1). S. 25 - 57. WARSAW: ARS POLONA-RUCH. ISSN 1730-6302
Full text not available from this repository.Abstract
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.
Item Type: | Journal Article | ||||||||||||
Creators: |
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URN: | urn:nbn:de:hbz:38-139730 | ||||||||||||
DOI: | 10.4064/cm7375-3-2018 | ||||||||||||
Journal or Publication Title: | Colloq. Math. | ||||||||||||
Volume: | 156 | ||||||||||||
Number: | 1 | ||||||||||||
Page Range: | S. 25 - 57 | ||||||||||||
Date: | 2019 | ||||||||||||
Publisher: | ARS POLONA-RUCH | ||||||||||||
Place of Publication: | WARSAW | ||||||||||||
ISSN: | 1730-6302 | ||||||||||||
Language: | English | ||||||||||||
Faculty: | Unspecified | ||||||||||||
Divisions: | Unspecified | ||||||||||||
Subjects: | no entry | ||||||||||||
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Refereed: | Yes | ||||||||||||
URI: | http://kups.ub.uni-koeln.de/id/eprint/13973 |
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