A Note on the Diophantine Equation x! plus A = y(2), II

In this note, we study the variant of the Brocard-Ramanujan Diophantine equation x! + A = y(2). In a previous note, we studied a system of three equations of the type Brocard-Ramanujan. Here, first we consider another pattern, precisely the cases A = 4k + 3, A = 4(k + 1)+ 3, A = 4(k + 2), where k epsilon N. We use an elementary approach to prove that these simultaneous equations have only two integral solutions. Then we show that the set of vertical bar A vertical bar is of asymptotic density zero.