Multiplicities of binary recurrences (Q2739247)

Let \(K\) be an algebraic number field of degree \(d\) and \(u_0\), \(u_1\) algebraic integers in \(K\), and \(\omega\in K^*\). Consider a nondegenerate binary recurrence \((u_n)_{n\geq 0}\) with companion polynomial \(f(X)=(X-\lambda)(X-\mu)\). Then, the authors prove: NEWLINENEWLINENEWLINETheorem. If \(\min\{|\lambda|, |\mu|\}>1\) and \(\max\{\text{H}(u_0), \text{H}(u_1)\}>c(d,f,\omega)\), where \(c(d,f,\omega)\) is an effectively computable constant, then the equation \(u_n=\omega\) has at most one solution. The main tool for the proof is linear forms in logarithms.