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

Abstract

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. 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 $0<r\le 10^6$, $p\ge 5$ 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.