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.

12 pages