Perfect powers in products of integers from a block of consecutive integers (II)

Abstract

(1) (m+ d1) . . . (m+ dt) = by in m, t, d1, . . . , dt, b, y and l. We always assume that the left hand side of equation (1) is divisible by a prime exceeding k. Consequently, there is an i with 1 ≤ i ≤ t such that m + di is divisible by an lth power of a prime exceeding k. Thus m+ di ≥ (k + 1) implying that m > k. Equation (1) with t = k and b = 1 is solved completely by Erdős and Selfridge [5] in 1975; a product of two or more consecutive positive integers is never a power. In fact, Erdős [4] proved in 1955 that for e > 0, equation (1) with b = 1 and