In our paper we initiate a systematic treatment for solving the title equation for bounded positive integer coefficients A, B and C. To illustrate our approach we explicitly solve the equation in integers x, y and n with |xy| > 1, n ≥ 3 for a collection of coefficients A, B, C. We first derive, for concrete values of A, B, C ≤ 100, a relatively small upper bound for n, provided that the equation under consideration has no solution with |xy| ≤ 1 (cf. Theorem 1). Then we give among others all the solutions (x, y, n) for C = 1, A, B ≤ 20 (cf. Theorem 3), and for A = C = 1, B ≤ 70 (cf. Theorem 4). Our method, which may, with some effort, be extended to larger values of A, B and C, combines a wide variety of techniques, classical and modern, in Diophantine analysis.

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