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

Nirvana Coppola; Mar Curcó-Iranzo; Maleeha Khawaja; Vandita Patel; Özge Ülkem

doi:10.1016/j.indag.2024.03.011

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

Nirvana Coppola, Mar Curcó-Iranzo, Maleeha Khawaja, Vandita Patel^*, Özge Ülkem

^*Corresponding author for this work

Research output: Contribution to journal › Article › Academic › peer-review

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=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≤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.

Original language	English
Pages (from-to)	500-515
Number of pages	16
Journal	Indagationes Mathematicae
Volume	35
Issue number	3
DOIs	https://doi.org/10.1016/j.indag.2024.03.011
Publication status	Published - May 2024

Bibliographical note

Publisher Copyright:
© 2024 The Author(s)

Funding

This project stemmed from the Women in Numbers Europe 4 workshop, which took place in August 2022 at Utrecht University. The authors are immensely appreciative towards the organisers: Ramla Abdellatif, Valentijn Karemaker, Ariane M\u00E9zard and Nirvana Coppola for hosting such an inspiring and productive workshop, and for all of their time committed towards such a noble endeavour. N. Coppola is supported by the NWO Vidi grant No. 639.032.613, New Diophantine Directions. M. Khawaja is supported by an EPSRC studentship from the University of Sheffield (EPSRC grant no. EP/T517835/1). \u00D6. \u00DClkem is supported by T\u00DCBITAK project no. 119F405. The authors are grateful to the referee for carefully reading the work and suggesting improvements. N. Coppola is supported by the NWO Vidi grant No. 639.032.613 , New Diophantine Directions. M. Khawaja is supported by an EPSRC studentship from the University of Sheffield (EPSRC grant no. EP/T517835/1 ). \u00D6. \u00DClkem is supported by T\u00DCBITAK project no. 119F405 .

Funders	Funder number
Universiteit Utrecht
Engineering and Physical Sciences Research Council
Nederlandse Organisatie voor Wetenschappelijk Onderzoek	639.032.613
University of Sheffield	EP/T517835/1
Türkiye Bilimsel ve Teknolojik Araştırma Kurumu	119F405

Keywords

Baker's Bounds
Exponential equation
Lehmer sequences
Primitive divisors
Thue equation

Access to Document

10.1016/j.indag.2024.03.011Licence: CC BY

1-s2.0-S0019357724000296-mainFinal published version, 348 KBLicence: CC BY

Cite this

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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. Arg{\'a}ez-Garc{\'i}a and the fourth author (perfect powers that are sums of cubes of a three, five and seven term arithmetic progression). Under the assumptions 06, 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.",

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AU - Ülkem, Özge

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AB - 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 06, 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.

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

Abstract

Bibliographical note

Funding

Keywords

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