On perfect powers that are sums of cubes of a seven term arithmetic progression

We prove that the equation (x − 3r) 3 + (x − 2r) 3 + (x − r) 3 + x 3 + (x + r) 3 + (x + 2r) 3 + (x + 3r) 3 = y p only has solutions which satisfy xy = 0 for 1 ≤ r ≤ 106 and p ≥ 5 prime. This article complements the work on the equations (x − r) 3 + x 3 + (x + r) 3 = y p in [2] and (x − 2r) 3 + (x − r) 3 + x 3 + (x + r) 3 + (x + 2r) 3 = y p in [1]. The methodology in this paper makes use of the Primitive Divisor Theorem due to Bilu, Hanrot and Voutier for a complete resolution of the Diophantine equation.