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+x3+(x+r)3+(x+2r)3+(x+3r)3+(x+4r)3=yp, which is a natural continuation of previous works carried out by A. Argáez-García 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≤106, 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.