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. Arg´ aez-Garc´ ıa 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.