D(−1)-quadruples and products of two primes

Abstract

.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+n(2) 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 +r(2),c = 1+s(2) and d = 1 +t(2), then it is known that r, s, t,b,c and d are not of the form p(k) or 2p(k), 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 p(2)v-q(2)w = 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 x(2) - (1+k(2))y(2) = k(2). 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+ t(2)