THE NON-EXISTENCE OF D(-1)-QUADRUPLES EXTENDING CERTAIN PAIRS IN IMAGINARY QUADRATIC RINGS

A D(n)-m-tuple, where n is a non-zero integer, is a set of m distinct elements in a commutative ring R such that the product of any two distinct elements plus n is a perfect square in R. In this paper, we prove that there does not exist a D(-1)-quadruple {a, b, c, d} in the ring Z[root-k], k >= 2 with positive integers a < b < 16a(2) - a -2 + 2 root k(8a(2) + 3a +1) and integers c and d satisfying d < 0 < c. By combining that result with [14, Theorem 1.1] we were able to obtain a general result on the non-existence of a D(-1)-quadruple {a, b, c, d} in Z[root- k] with integers a, b, c, d satisfying a < b <= 8a - 3. Furthermore, for a non-negative integer i and a positive integer j, we apply the obtained results in proving of the non-existence of D(-1)-quadruples containing powers of primes p(i), q(j) with an arbitrary different primes p and q.