Solutions of Diophantine equations in ordered fields

Subject Areas : Number theory

M. A. P. Cabral ¹ , F. C. Marques ² , D. P. Pombo Jr. ³

1 - Institute of Mathematics, Federal University of Rio de Janeiro, Rio de Janeiro - RJ, Brazil
2 - Institute of Mathematics and Statistics, Fluminense Federal University, Niter\'oi - RJ, Brazil
3 - Institute of Mathematics and Statistics, Fluminense Federal University, Niter\'oi - RJ, Brazil

Received: 2025-01-06 Accepted : 2025-05-14 Published : 2025-05-20

Keywords: Diophantine equations, ordered fields, computer algebra system,

Abstract :

For each integer $n \geqslant 2$, solutions of the Diophantine equations \[x_1^3 + x_2^3 + \cdots + x_{2n-1}^3 + x_{2n}^3 =y_1^3 + y_2^3 + \cdots + y_{2n-1}^3 + y_{2n}^3\] and \[x_1^4 + x_2^4 + \cdots + x_{2n-1}^4 + x_{2n}^4 =y_1^4 + y_2^4 + \cdots + y_{2n-1}^4 + y_{2n}^4\] in an arbitrary ordered field are exhibited.‎

References:

[1] N. Bourbaki, Algèbre, Chapitre 6, Deuxième édition, Actualités Scientifiques et Industrielles 1179, Hermann, Paris, 1964.

[2] G. H. Hardy, E. M. Wright, An Introduction to the Theory of Numbers, Fifth edition, Oxford University Press, Oxford, 1979.

[3] F. C. Marques, D. P. Pombo Jr., Rational solutions of certain Diophantine equations, JP J. Math. Sci. 31 (2022), 1-18.

[4] P. Swinnerton-Dyer, A solution of A4+ B4= C4+ D4, J. London Math. Soc. 18 (1943), 2-4.