Euler indicators of binary recurrence sequences. (Q1396455)

Let \(\varphi(n)\) denote Euler's totient function. Let \((u_n)_{n\geq 0}\) and \((v_n)_{n\geq 0}\) be binary linear recurrences, that is, \[ u_{n+2}= r_1 u_{n+1}+ s_1 u_n,\quad v_{n+2}= r_2 v_{n+1}+ s_2 v_n \] for \(n\geq 0\). Let the initial terms \(u_0\), \(u_1\), \(v_0\), \(v_1\) as well as the coefficients \(r_1\), \(s_1\), \(r_2\), \(s_2\) be given integers. Suppose also that \(r^2_1+ 4s_1> 0\), \(r^2_2+ 4s_2> 0\), \(v_0= 2\), \(v_1= r_2\). Under certain additional assumptions, the author proves that the Diophantine equation \[ \varphi(| au_m|)=| bv_n| \] has at most finitely many solutions. The proof uses estimations of linear forms in logarithms of algebraic numbers. As a corollary, the author obtains all solutions of the equations \[ \varphi(L_m)= L_n,\quad \varphi(F_m)= L_n, \] where \(F_n\), \(L_n\) denote the \(n\)th Fibonacci and Lucas numbers, respectively.