The multiplicity for a class of second order recurrences (Q1200706)

Let \(a_ 1,a_ 2\) be coprime non-zero integers with \(a_ 2\neq\pm1\), and \(U=\{u_ m\}_{m=0}^ \infty\) be an integer sequence satisfying \(u_ 0=0\), \(u_ 1=1\), \(u_{m+2}=a_ 1u_{m+1}+a_ 2u_ m\), \(m\geq 0\). Then \(u_ m=(\varepsilon^ m-\bar\varepsilon^ m)/(\varepsilon- \bar\varepsilon)\), \(m\geq 0\), where \(\varepsilon=(a_ 1+\sqrt{D})/2\), \(\bar\varepsilon=(a_ 1-\sqrt{D})/2\), \(D=a_ 1^ 2+4a_ 2\). For any positive integer \(k\), let \(N(K)\) denote the number of positive integers \(m\) such that \(| u_ m|=k\). In [Compos. Math. 40, 251-267 (1980; Zbl 0396.10005)], \textit{F. Beukers} proved that \(N(k)\leq 3\). In this note we shall prove the following: Theorem. When \(\max(| D|,| a_ 2|)>M=\exp\exp\exp 1000\), the following statements hold: (i) If \(a_ 1=\pm1\) or \(a_ 1^ 2+a_ 2=\pm1\), then \(N(1)=2\). (ii) If \(a_ 1^ 2+2a_ 2=\pm1\), then \(N(| u_ 2|)=2\). (iii) Excepting the above cases, \(N(k)\leq 1\).