The exponential diophantine equation AX² + BY² = λk^z and its applications

Let A, B ∈ N with A > 1, B > 1 and gcd(A, B) = 1, k ≥ 2 be an integer coprime with AB, and let λ ∈ {1, 2, 4} be such that if λ = 4, then A ≠ 4 and B ≠ 4; and if k is even, then λ = 4. In this paper, we shall describe all solutions of the equation AX² + BY² = λk^Z, X, Y, Z ∈ Z, gcd(X, Y) = 1, Z > 0 with X|*A or Y |*B, where the symbol X|*A means that every prime divisor of X divides A. Then, using this result, we give some more general results on the number of solutions of the equation la^x + mb^y = λc^z, x > 1, y > 1, z >1. In addition, using Cao's resulton Pell equation, we obtain some improvement of Terai's results on the equations a^x + 2 = c^z, a^x + 4 = c^z and a^x + 2^y = c^z.