Divisibility properties of binary recurrent sequences
Let \(P\) be any set of primes and \((u_n)_{n\geq 0}\) be a nondegenerate binary recurrence whose \(n\)th term is given by \(u_n= c\alpha^n+ d\beta^n\). The author proves: If \(c/d\) and \(\alpha/\beta\) are multiplicatively independent, then there exist two effectively computable constants \(c_1\) and \(c_2\) depending on the sequence \((u_n)_{n\geq 0}\) and the set \(P\) such that whenever \(m\neq n\) are two integers with \(u_m/u_n\) an \(S\)-integer and \(n> c_1\), then \(m> c_2 \frac{n^2}{\log n}\). There is another statement about the multiplicatively dependent case.
scientific article; zbMATH DE number 1766476