On Perfect Powers That Are Sums of Cubes of a Three Term Arithmetic Progression

摘要

Using only elementary arguments, Cassels and Uchiyama (independently) determined all squares that are sums of three consecutive cubes. Zhongfeng Zhang extended this result and determined all perfect powers that are sums of three consecutive cubes. Recently, the equation (x - r)~k + x~k + {x + r)~k has been studied for k = 4 by Zhongfeng Zhang and for k = 2 by Koutsianas and Patel. In this paper, we complement the work of Cassels, Koutsianas, Patel and Zhang by considering the case when k = 3 and showing that the equation (x - r)~3 + x~3 + (x + r)~3 = y~n with n ≥ 5 a prime and 0 < r ≤ 10~6 only has trivial solutions (x, y, n) which satisfy xy = 0.