Comptes Rendus
Article de recherche - Théorie des nombres
Note on a symmetric Diophantine equation
[Note sur une équation diophantienne symétrique]
Tho Nguyen Xuan1
1 Faculty of Mathematics and Informatics, Hanoi University of Science and Technology, Hanoi, Vietnam
Comptes Rendus. Mathématique, Volume 363 (2025), pp. 985-988

Using an elementary argument, we show that for all rational numbers $\alpha $ such that neither $\alpha $ nor $3\alpha $ is a rational square, the equation

\[ x^4-4\alpha x^2-4\alpha y^2+y^4=-6\alpha ^2 \]

has no rational solutions. This answers Hindes’ two questions and generalizes his theorem (Theorem 1.1) in “Rational points on certain families of symmetric equations”, Int. J. Number Theory 11 (2015), no. 6, pp. 1821–1838.

En utilisant un argument élémentaire, nous montrons que pour tous les nombres rationnels $\alpha $ tels que ni $\alpha $ ni $3\alpha $ n’est un carré rationnel, l’équation

\[ x^4-4\alpha x^2-4\alpha y^2+y^4=-6\alpha ^2 \]

n’a pas de solutions rationnelles. Ceci répond aux deux questions de Hindes et généralise son théorème (Theorem 1.1) dans “Rational points on certain families of symmetric equations”, Int. J. Number Theory 11 (2015), no. 6, pp. 1821–1838.

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DOI : 10.5802/crmath.785
Classification : 14G05, 14G12
Keywords: Arithmetic geometry, Diophantine equations, rational points
Mots-clés : Géométrie arithmétique, équations diophantiennes, points rationnels

Tho Nguyen Xuan 1

1 Faculty of Mathematics and Informatics, Hanoi University of Science and Technology, Hanoi, Vietnam
Licence : CC-BY 4.0
Droits d'auteur : Les auteurs conservent leurs droits 
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Tho Nguyen Xuan. Note on a symmetric Diophantine equation. Comptes Rendus. Mathématique, Volume 363 (2025), pp. 985-988. doi: 10.5802/crmath.785

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