Quadruples $(a,b,c,d)$ of positive integers $a<b<c<d$ with the property that the product of any two of them is one more than a perfect square are studied. Improved lower and upper bounds for the entries $b$ and $c$ are established. As an application of these results, a bound for the number of such quadruples is obtained.
Nicolae Ciprian Bonciocat. Mihai Cipu. Maurice Mignotte. "On $D(-1)$-Quadruples." Publ. Mat. 56 (2) 279 - 304, 2012.