LAUSR

Abstract We prove that the equation ( x − 3 r ) 3 + ( x − 2 r ) 3 + ( x − r ) 3 + x 3 + ( x + r ) 3 + ( x + 2 r ) 3 + ( x + 3 r ) 3 = y p only has solutions which satisfy x y = 0 for 1 ≤ r ≤ 10 6 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 − 2 r ) 3 + ( x − r ) 3 + x 3 + ( x + r ) 3 + ( x + 2 r ) 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.