Copy For Citation
Coppola N., Curcó-Iranzo M., Khawaja M., Patel V., Ülkem Ö.
Indagationes Mathematicae, vol.35, no.3, pp.500-515, 2024 (SCI-Expanded)
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Publication Type:
Article / Article
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Volume:
35
Issue:
3
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Publication Date:
2024
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Doi Number:
10.1016/j.indag.2024.03.011
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Journal Name:
Indagationes Mathematicae
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Journal Indexes:
Science Citation Index Expanded (SCI-EXPANDED), Scopus, MathSciNet, zbMATH, DIALNET
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Page Numbers:
pp.500-515
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Keywords:
Baker's Bounds, Exponential equation, Lehmer sequences, Primitive divisors, Thue equation
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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.