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

We study the equation (x-4r)(3)+(x-3r)(3)+(x-2r)(3)+(x-r)(3)+x(3)+(x+r)(3)+(x+2r)(3)+(x+3r)(3)+(x+4r)(3 )= y(p), which is a natural continuation of previous works carried out by A. Argaez-Garcia and the fourth author (perfect powers that are sums of cubes of a three, five and seven term arithmetic progression). Under the assumptions 0 < r <= 10(6), p >= 5 a prime and gcd(x,r) = 1, we show that solutions must satisfy xy = 0. Moreover, we study the equation for prime exponents 2 and 3 in greater detail. Under the assumptions r>0 a positive integer and gcd(x,r) = 1 we show that there are infinitely many solutions for p = 2 and p = 3 via explicit constructions using integral points on elliptic curves. We use an amalgamation of methods in computational and algebraic number theory to overcome the increased computational challenge. Most notable is a significant computational efficiency obtained through appealing to Bilu, Hanrot and Voutier's Primitive Divisor Theorem and the method of Chabauty, as well as employing a Thue equation solver earlier on. (c) 2024 The Author(s). Published by Elsevier B.V. on behalf of Royal Dutch Mathematical Society (KWG).This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/4.0/).