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

Coppola N., Curcó-Iranzo M., Khawaja M., Patel V., Ülkem Ö.

Indagationes Mathematicae, vol.35, no.3, pp.500-515, 2024 (SCI-Expanded, Scopus)

  • Publication Type: Article / Article
  • Volume: 35 Issue: 3
  • Publication Date: 2024
  • Doi Number: 10.1016/j.indag.2024.03.011
  • Journal Name: Indagationes Mathematicae
  • Journal Indexes: Science Citation Index Expanded (SCI-EXPANDED), Scopus, MathSciNet, zbMATH, DIALNET
  • Page Numbers: pp.500-515
  • Keywords: Baker's Bounds, Exponential equation, Lehmer sequences, Primitive divisors, Thue equation
  • Galatasaray University Affiliated: Yes

Abstract

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 00 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.