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Ciolan, Alexandru and Moree, Pieter (2019). BROWKIN'S DISCRIMINATOR CONJECTURE. Colloq. Math., 156 (1). S. 25 - 57. WARSAW: ARS POLONA-RUCH. ISSN 1730-6302

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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:	Creators Email ORCID ORCID Put Code Ciolan, Alexandru UNSPECIFIED UNSPECIFIED UNSPECIFIED Moree, Pieter UNSPECIFIED UNSPECIFIED UNSPECIFIED
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
Uncontrolled Keywords:	Keywords Language Mathematics Multiple languages
Refereed:	Yes
URI:	http://kups.ub.uni-koeln.de/id/eprint/13973