On the greatest prime factor and divisibility properties of linear recursive sequences
Let K be an algebraic number field and \({\mathcal O}_ K\) its ring of integers. Let \(a_ 1,a_ 2\in K\) and \(\lambda\),mu\(\in {\mathcal O}_ K\), all non-zero. The author considers the numbers \(x_ m=a_ 1\lambda^ m+a_ 2\mu^ m\) \((m=0,1,2,...)\) and studies the problem for which n,m we have \(x_ n| x_ m\). In earlier papers by Parnami, Shorey and Steward this problem was considered in the case when the \(u_ m\) are rational integers. Presently, the author drops this constraint and manages to find effective lower bounds for m-2n and for the largest (in a suitable sense) prime factor in \(x_ m/(x_ m,x_ n)\).
scientific article; zbMATH DE number 4154524
scientific article