Exploring 8<sup>x</sup> + n<sup>y</sup> = z<sup>2</sup> through Associated Elliptic Curves
Issued Date
2025-01-01
Resource Type
ISSN
18140424
eISSN
18140432
Scopus ID
2-s2.0-85204962387
Journal Title
International Journal of Mathematics and Computer Science
Volume
20
Issue
1
Start Page
247
End Page
254
Rights Holder(s)
SCOPUS
Bibliographic Citation
International Journal of Mathematics and Computer Science Vol.20 No.1 (2025) , 247-254
Suggested Citation
Panraksa C. Exploring 8<sup>x</sup> + n<sup>y</sup> = z<sup>2</sup> through Associated Elliptic Curves. International Journal of Mathematics and Computer Science Vol.20 No.1 (2025) , 247-254. 254. doi:10.69793/ijmcs/01.2025/chatchawan Retrieved from: https://repository.li.mahidol.ac.th/handle/20.500.14594/102815
Title
Exploring 8<sup>x</sup> + n<sup>y</sup> = z<sup>2</sup> through Associated Elliptic Curves
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Abstract
This paper investigates the exponential Diophantine equation 8x+ ny = z2, where n > 1 is an odd positive integer. We characterize solutions for the base cases (x = 0 or y = 0) and describe, based on implications of Bennett and Skinner’s theorem, that no solutions exist for y > 2 in certain cases. For y = 1 and y = 2, we employ elliptic curve methods, focusing on the equations z2 = t3+n and z2 = t3+n2, where t = 2x. This work generalizes known results for specific cases and provides insights into this class of Diophantine equations and their associated elliptic curves.