The shuffle variant of Terai's conjecture on exponential Diophantine equations

Let p, q and r be positive integers with p, q, r >= 2, and let a, b and c be pair-wise relatively prime positive integers such that a(p) + b(q) = c(r). Terai's conjecture states that apart from a handful of exceptions, the exponential Diophantine equation a(x) + b(y) = c(z) in positive integers x, y and z, has the unique solution (x, y, z) = (p, q, r). In this paper we consider a similar problem (which we call the shuffle variant of Terai's problem). Our problem states that apart from a handful of exceptions, the exponential Diophantine equation c(x) + b(y) = a(z) in positive integers x, y and z, has the unique solution (x, y, z) = (1,1,p) if q = r = 2 and c = b + 1, and no solutions otherwise. We establish several results on our problem by the theory of linear forms in two archimedean and non-archimedean logarithms with various elementary techniques. In particular we prove that the shuffle variant of Terai's problem is true if q = r = 2 and c = b + 1.