Variations on the Brocard–Ramanujan equation

We study Diophantine equations of the shape y2=BUn+A, where A and B are fixed integers, Un=f(1)f(2)⋯f(n), and f:N+→N+ is an increasing function. We prove, in particular, several results concerning the existence of pairs (A,B) (and fʼs) such that the above equations have at least four solutions in positive integers y and n. One of the main ingredient in the proof is description of linear polynomials which, for given integers a, b, c, d take values which are squares of integers. We also collect some numerical computations, observations, questions and conjectures related to the subject.