General Ramanujan-type diophantine equations and their complete solution (Q1907839)

The author revisits the famous Ramanujan equation \(x^2+ 7= 2^{n+ 2}\). He observes that its solution is essentially reduced to proving that the only positive integers \(n\) for which \(A_n= -1\) are \(n= 3, 5, 13\); here \(A_n\) is the recurrence sequence defined by \(A_1= 1\), \(A_2= 1\), \(A_n= A_{n- 1}- 2A_{n- 2}\). The author generalizes the above problem to that of finding all possible repetitions occurring in the values of \(A_n\). Associated to \(A_n\) is the recurrence sequence \(B_n\) defined by \(B_1= 1\), \(B_2= -3\), \(B_n= B_{n- 1}- 2B_{n- 2}\). The following identity holds: \((*)\) \(B^2_n+ 7 A^2_n= 2^{n+ 2}\) and the author shows by elementary means twelve interesting properties of the sequences \(A_n\) and \(B_n\). With the aid of them he proves then that the only repeated values in \(A_n\) are: 1 for \(n= 1,2\); \(-1\) for \(n= 3, 5, 13\) and \(-3\) for \(n= 4,8\). Also, the only repeated values in \(B_n\) are 1 for \(n= 1, 4\) and \(- 5\) for \(n= 3, 9\). Remark: In p. 161 the author says that ``we will find all solutions of equation \((*)\). Since \((*)\) is an identity, this assertion puzzled the reviewer; it took some time for him to understand that by ``solution to \((*)\)'' the author means those \((m, n)\) with \(m\neq n\) for which either \(A_m= A_n\) or \(B_m= B_n\).