On the extendibility of certain D(-1)-pairs in imaginary quadratic rings

Let R be a commutative ring with unity 1. A set of m different non-zero elements in R such that the product of any two distinct elements decreased by 1 is a perfect square in R is called a D(-1)-m-tuple in R. In the ring Z[root-k], with an integer k >= 2, we consider the D(-1)-pairs {a, 2(i)p(j)}, where i is an element of{0,1}, a, j are positive integers, p is an odd prime, gcd(a, 2(i)p(j)) = 1 and a < 2(i)p(j). We prove that there does not exist a D(-1)-quadruple of the form {a, 2(i)p(j), c, d} in Z[root-k] in the following cases: k does not divide 2(i)p(j) - a; k divides 2(i)p(j) - a and (2(i)p(j) - a)/k is a prime; k = 2(i)p(j) - a and a > 1.