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.

中文翻译：

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关于作为七项等差数列的立方和的完美幂

摘要 我们证明方程 ( x − 3 r ) 3 + ( x − 2 r ) 3 + ( x − r ) 3 + x 3 + ( x + r ) 3 + ( x + 2 r ) 3 + ( x + 3 r ) 3 = yp 只有满足 xy = 0 且 1 ≤ r ≤ 10 6 且 p ≥ 5 素数的解。本文补充了 [2] 中方程 ( x − r ) 3 + x 3 + ( x + r ) 3 = yp 和 ( x − 2 r ) 3 + ( x − r ) 3 + x 3 + ( x + r ) 3 + ( x + 2 r ) 3 = yp 在 [1] 中。本文中的方法利用 Bilu、Hanrot 和 Voutier 的原始除数定理来完整解析丢番图方程。