 Some Results on Divisibility SequencesNorbert Jensen A chapter in Applications of Fibonacci Numbers, 1990, pp 181-189 from Springer Abstract: Abstract Let g be a sequence of rational integers, g is called a divisibility sequence (DS) iff (1.1) $$n|m \to g\left( n \right)|g\left( m \right)$$ holds for all positive integers n, m. (This concept has been introduced by Ward in [8], [9]. Ward generalized a concept used by Hall in [2] where he considered DS satisfying a linear recurrence relation of order k with the coefficients being rational integers. Hall’s work has been inspired by works of other authors for special cases, e.g.: E. Lucas, D. H. Lehmer, T. A. Pierce.) $$g is called a strong divisiblity sequence \left( {SDS} \right), iff even$$ (1.2) $$\left( {g\left( n \right),g\left( m \right)} \right) = g\left( {\left( {n,m} \right)} \right)$$ holds for all positive integers n,m. (This term has first been used by Kimberling in [4]. Here g satisfies a linear recurrence relation of arbitrary order. The general definition is given by Kimberling in [5]. However, such sequences have already been considered by Ward in [8], [9], [12], 1936–39.) The Fibonacci- Sequence defined by u1 = 1, u2 = 1, u n +2 = u n +1 + u n on the set of positive integers ℕ is a special SDS, as can be seen from Theorem 6.1, see Carmichael [1]. Date: 1990 There are no downloads for this item, see the EconPapers FAQ for hints about obtaining it. Related works: Export reference: BibTeX RIS (EndNote, ProCite, RefMan) HTML/Text Persistent link: https://EconPapers.repec.org/RePEc:spr:sprchp:978-94-009-1910-5_20 Ordering information: This item can be ordered from DOI: 10.1007/978-94-009-1910-5_20 Access Statistics for this chapter More chapters in Springer Books from Springer |
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