Diophantine equations with products of consecutive values of a quadratic polynomial

Abstract

Let a, b, c, d be given nonnegative integers with a , d ⩾ 1 . Using Chebyshevʼs inequalities for the function π ( x ) and some results concerning arithmetic progressions of prime numbers, we study the Diophantine equation ∏ k = 1 n ( a k 2 + b k + c ) = d y l , gcd ( a , b , c ) = 1 , l ⩾ 2 , where a x 2 + b x + 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.