dc.contributor.author Turner, John C. dc.contributor.author Rogers, Bill dc.date.accessioned 2015-01-18T22:21:43Z dc.date.available 2013 dc.date.available 2015-01-18T22:21:43Z dc.date.issued 2013 dc.identifier.citation Turner, 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.issn 1787-5021 dc.identifier.uri https://hdl.handle.net/10289/9070 dc.description.abstract In 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.extent 235 - 254 dc.format.mimetype application/pdf dc.language eng dc.language.iso en dc.rights This article has been published in the journal: Annales Mathematicae et Informaticae. Used with permission. dc.subject Cycle-number dc.subject Cycle-number triangle dc.subject Prime cycle-numbers dc.title A representation of the natural numbers by means of cycle-numbers, with consequences in number theory dc.type Journal Article dc.relation.isPartOf Annales Mathematicae et Informaticae pubs.begin-page 235 pubs.elements-id 84351 pubs.end-page 254 pubs.volume 41 dc.identifier.eissn 1787-6117
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