Abstract
We give a new proof that all rational points on \(y^2=x^6+x^2+1\) are \(\pm \infty \), \((0,\pm 1),\,(\pm \dfrac{1}{2},\pm \dfrac{9}{8})\). Our approach combines the two descent map on elliptic curves with the elliptic curve Chabauty method over certain quartic number fields.
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This author is partially supported by Vietnam National Foundation for Science and Technology Development (NAFOSTED) [grant number 101.04-2019.314].
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Communicated by B. Sury.
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Tho, N.X. The equation \(y^2=x^6+x^2+1\) revisited. Indian J Pure Appl Math 54, 760–765 (2023). https://doi.org/10.1007/s13226-022-00294-x
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DOI: https://doi.org/10.1007/s13226-022-00294-x