D(−1)-quadruples and Products of Two Primes

A D(−1)-quadruple is a set of positive integers {a, b, c, d}, with a < b < c < d, such that the product of any two elements from this set is of the form 1+n2 for some integer n. Dujella and Fuchs showed that any such D(−1)-quadruple satisfies a = 1. The D(−1) conjecture states that there is no D(−1)-quadruple. If b = 1+ r2, c = 1+ s2 and d = 1+ t2, then it is known that r, s, t, b, c and d are not of the form p or 2p, where p is an odd prime and k is a positive integer. In the case of two primes, we prove that if r = pq and v and w are integers such that p2v−q2w = 1, then 4vw − 1 > r. A particular instance yields the result that if r = p(p + 2) is a product of twin primes, where p ≡ 1 (mod 4), then the D(−1)-pair {1, 1+r2} cannot be extended to a D(−1)-quadruple. Dujella’s conjecture states that there is at most one solution (x, y) in positive integers with y < k− 1 to the diophantine equation x2 − (1+ k2)y2 = k2. We show that the Dujella conjecture is true when k is a product of two odd primes. As a consequence it follows that if t is a product of two odd primes, then there is no D(−1)-quadruple {1, b, c, d} with d = 1 + t2.