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dc.contributor.authorTurner, John C.
dc.contributor.authorRogers, Bill
dc.date.accessioned2015-01-18T22:21:43Z
dc.date.available2013
dc.date.available2015-01-18T22:21:43Z
dc.date.issued2013
dc.identifier.citationTurner, J. C., & Rogers, B. (2013). A representation of the natural numbers by means of cycle-numbers, with consequences in number theory. Annales Mathematicae Et Informaticae, 41, 235–254.en
dc.identifier.issn1787-5021
dc.identifier.urihttps://hdl.handle.net/10289/9070
dc.description.abstractIn this paper we give rules for creating a number triangle T in a manner analogous to that for producing Pascal's arithmetic triangle; but all of its elements belong to {0, 1}, and cycling of its rows is involved in the creation. The method of construction of any one row of T from its preceding rows will be defined, and that, together with starting and boundary conditions, will suffice to define the whole triangle, by sequential continuation. We shall use this triangle in order to define the so-called cycle-numbers, which can be mapped to the natural numbers. T will be called the 'cyclenumber triangle'. First we shall give some theorems about relationships between the cyclenumbers and the natural numbers, and discuss the cycling of patterns within the triangle's rows and diagonals. We then begin a study of figures (i.e. (0,1)- patterns, found on lines, triangles and squares, etc.) within T. In particular, we shall seek relationships which tell us something about the prime numbers. For our later studies, we turn the triangle onto its side and work with a doubly-infinite matrix C. We shall find that a great deal of cycling of figures occurs within T and C, and we exploit this fact whenever we can. The phenomenon of cycling patterns leads us to muse upon a 'music of the integers', indeed a 'symphony of the integers', being played out on the cycle-number triangle or on C. Like Pythagoras and his 'music of the spheres', we may well be the only persons capable of hearing it!.
dc.format.extent235 - 254
dc.format.mimetypeapplication/pdf
dc.languageeng
dc.language.isoen
dc.rightsThis article has been published in the journal: Annales Mathematicae et Informaticae. Used with permission.
dc.subjectCycle-number
dc.subjectCycle-number triangle
dc.subjectPrime cycle-numbers
dc.titleA representation of the natural numbers by means of cycle-numbers, with consequences in number theory
dc.typeJournal Article
dc.relation.isPartOfAnnales Mathematicae et Informaticae
pubs.begin-page235
pubs.elements-id84351
pubs.end-page254
pubs.volume41
dc.identifier.eissn1787-6117


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