Some Results on Divisibility Sequences

Abstract

Let g be a sequence of rational integers, g is called a divisibility sequence (DS) iff $$n|m \to g\left( n \right)|g\left( m \right)$$ (1.1) 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.) $$g is called a strong divisiblity sequence \left( {SDS} \right), iff even$$ $$\left( {g\left( n \right),g\left( m \right)} \right) = g\left( {\left( {n,m} \right)} \right)$$ (1.2) 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].