6 research outputs found

    Artin's primitive root conjecture -a survey -

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    This is an expanded version of a write-up of a talk given in the fall of 2000 in Oberwolfach. A large part of it is intended to be understandable by non-number theorists with a mathematical background. The talk covered some of the history, results and ideas connected with Artin's celebrated primitive root conjecture dating from 1927. In the update several new results established after 2000 are also discussed.Comment: 87 pages, 512 references, to appear in Integer

    Cycles of Sums of Integers

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    We study the period of the linear map T:Zmn→Zmn:(a0,…,an−1)↦(a0+a1,…,an−1+a0)T:\textbf{Z}_m^n\rightarrow \textbf{Z}_m^n:(a_0,\dots,a_{n-1})\mapsto(a_0+a_1,\dots,a_{n-1}+a_0) as a function of mm and nn, where Zm\textbf{Z}_m stands for the ring of integers modulo mm. This map being a variant of the Ducci map, several known results are adapted in the context of TT. The main theorem of this paper states that the period modulo mm can be deduced from the prime factorization of mm and the periods of its prime factors. We give some other interesting properties.Comment: Submitted March 8 201

    Around Pelikan's conjecture on very odd sequences

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    Very odd sequences were introduced in 1973 by J. Pelikan who conjectured that there were none of length >=5. This conjecture was disproved by MacWilliams and Odlyzko in 1977 who proved there are in fact many very odd sequences. We give connections of these sequences with duadic codes, cyclic difference sets, levels (Stufen) of cyclotomic fields, and derive some new asymptotic results on their lengths and on S(n), which denotes the number of very odd sequences of length n.Comment: 21 pages, two tables. Revised version with improved presentation and correction of some typos and minor errors that will appear in Manuscripta Mathematic
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