<div class="csl-bib-body">
<div class="csl-entry">Drmota, M., Müllner, C., & Spiegelhofer, L. (2022, March 17). <i>Primes as sums of Fibonacci numbers</i> [Presentation]. Colloquium de l’IDP, France. http://hdl.handle.net/20.500.12708/153018</div>
</div>
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dc.identifier.uri
http://hdl.handle.net/20.500.12708/153018
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dc.description
The purpose of this paper is to discuss the relationship between prime numbers and sums of Fibonacci numbers. One of our main results says that for every sufficiently large integer k there exists a prime number that can be represented as the sum of k different and non-consecutive Fibonacci numbers. This property is closely related to, and based on, a prime number theorem for certain morphic sequences. The proof of such a prime number theorem, combined with a corresponding local result, is the central contribution of this paper, from which we derive the result stated in the beginning. Problems of this type have been discussed intensively in the context of the base-q expansion. The Gelfond problems (1968/1969), and the Sarnak conjecture, were the driving forces of this development. Mauduit and Rivat resolved the question on the sum of digits of prime numbers (2010) and the sum of digits of squares (2009), thus leaving open only part of the third Gelfond problem. Later the second author (2017) proved Sarnak's conjecture for all automatic sequences, which are based on the q-ary expansion of integers, and which generalize the sum-of-digits function in base q considerably. In order to obtain corresponding results for Fibonacci numbers, we have to extend Mauduit and Rivat's method considerably. In fact, we are departing significantly from this method, proving the statement that exp(2πiϑz(n)) has \emph{level of distribution} 1 (here z(n) is the number of Fibonacci numbers needed to write n as their sum). This latter result forms an essential part of our treatment of the occurring sums of type I and II and uses Gowers norms related to z(n) as a central technical tool. The appearance of Gowers norms in our method is intimately tied to the iterated application of a new generalization of van der Corput's inequality.
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dc.language.iso
en
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dc.subject
Fibonacci numbers
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dc.subject
Prime number theorem
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dc.subject
Level of distribution
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dc.subject
Sum-of-digits function,
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dc.subject
Zeckendorf expansion
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dc.title
Primes as sums of Fibonacci numbers
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dc.type
Presentation
en
dc.type
Vortrag
de
dc.contributor.affiliation
Montanuniversität Leoben, Austria
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dc.type.category
Presentation
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tuw.publication.invited
invited
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tuw.researchTopic.id
C4
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tuw.researchTopic.name
Mathematical and Algorithmic Foundations
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tuw.researchTopic.value
100
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tuw.publication.orgunit
E104-05 - Forschungsbereich Kombinatorik und Algorithmen
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tuw.author.orcid
0000-0002-2984-6005
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tuw.author.orcid
0000-0003-3552-603X
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tuw.event.name
Colloquium de l'IDP
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tuw.event.startdate
17-03-2022
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tuw.event.enddate
17-03-2022
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tuw.event.online
On Site
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tuw.event.type
Event for scientific audience
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tuw.event.country
FR
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tuw.event.presenter
Drmota, Michael
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wb.sciencebranch
Informatik
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wb.sciencebranch
Mathematik
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wb.sciencebranch.oefos
1020
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wb.sciencebranch.oefos
1010
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wb.sciencebranch.value
5
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wb.sciencebranch.value
95
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item.openairetype
conference presentation
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item.fulltext
no Fulltext
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item.grantfulltext
none
-
item.languageiso639-1
en
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item.openairecristype
http://purl.org/coar/resource_type/R60J-J5BD
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item.cerifentitytype
Publications
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crisitem.author.dept
E104 - Institut für Diskrete Mathematik und Geometrie
-
crisitem.author.dept
E104-05 - Forschungsbereich Kombinatorik und Algorithmen
-
crisitem.author.dept
E104 - Institut für Diskrete Mathematik und Geometrie
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crisitem.author.orcid
0000-0002-2984-6005
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crisitem.author.parentorg
E100 - Fakultät für Mathematik und Geoinformation
-
crisitem.author.parentorg
E104 - Institut für Diskrete Mathematik und Geometrie
