On diophantine equations of the form ${({\boldsymbol x}-{\boldsymbol a}_{\bf 1})({\boldsymbol x}-{\boldsymbol a}_{\bf 2}) \ldots ({\boldsymbol x}-{\boldsymbol a}_{\boldsymbol k}) + {\boldsymbol r} = {\boldsymbol y}^{ \boldsymbol n}}$

Abstract

Erdős and Selfridge [3] proved that a product of consecutive integers can never be a perfect power. That is, the equation x(x + 1)(x + 2)...(x + (m − 1)) = y n has no solutions in positive integers x,m,n where m, n > 1 and y ∈ Q. We consider the equation $$ (x-a_1)(x-a_2) \ldots (x-a_k) + r = y^n $$ where 0 ≤ a 1 < a 2 < ⋯ < a k are integers and, with r ∈ Q, n ≥ 3 and we prove a finiteness theorem for the number of solutions x in Z, y in Q. Following that, we show that, more interestingly, for every nonzero integer n > 2 and for any nonzero integer r which is not a perfect n-th power for which the equation admits solutions, k is bounded by an effective bound.