On the diophantine equation D(1)x(2)+D-2=k(n)

Let D, D-1, D-2, k be positive integers such that D = D-1 D-2, D-1 > 1, D-2 > 1, k > 1 and gcd(D-1, D-2) = gcd(D, k) = 1. Let w(k) be the number of distinct prime factors of k. Further, let N(D-1, D-2, k) be the number of positive integer solutions (x, n) of the equation D(1)x(2) + D-2 = k(n). In this paper, we prove that if 2 inverted iota k and max(D-1, D-2) > exp exp exp 105, then N(D-1, D-2, k) less than or equal to 2(w(k)-1) + 1 or 2(w(k)-1) according as the triple (D-1, D-2, k) is exceptional or not. The above upper bound is the best possible if k is a prime.