The equation $$y^2=x^6+x^2+1$$ y 2 = x 6 + x 2 + 1 revisited

Nguyen Xuan Tho ()
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Nguyen Xuan Tho: Hanoi University of Science and Technology

Indian Journal of Pure and Applied Mathematics, 2023, vol. 54, issue 3, 760-765

Abstract: Abstract We give a new proof that all rational points on $$y^2=x^6+x^2+1$$ y 2 = x 6 + x 2 + 1 are $$\pm \infty $$ ± ∞ , $$(0,\pm 1),\,(\pm \dfrac{1}{2},\pm \dfrac{9}{8})$$ ( 0 , ± 1 ) , ( ± 1 2 , ± 9 8 ) . Our approach combines the two descent map on elliptic curves with the elliptic curve Chabauty method over certain quartic number fields.

Keywords: Diophantine equation; Elliptic curve Chabauty; Rational points; Primary: 14G05; Secondary: 11G05 (search for similar items in EconPapers)
Date: 2023
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DOI: 10.1007/s13226-022-00294-x

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