Diophantine equations with products of consecutive values of a quadratic polynomial

Let a, b, c, d   be given nonnegative integers with a,d⩾1a,d⩾1. Using Chebyshevʼs inequalities for the function π(x)π(x) and some results concerning arithmetic progressions of prime numbers, we study the Diophantine equation∏k=1n(ak2+bk+c)=dyl,gcd(a,b,c)=1,l⩾2, where ax2+bx+cax2+bx+c is an irreducible quadratic polynomial. We provide a computable sharp upper bound to n. Using this bound, we entirely prove some conjectures due to Amdeberhan, Medina and Moll (2008) [1]. Moreover, we obtain all the positive integer solutions of some related equations.