Right triangles with algebraic sides and elliptic curves over number fields

Rok, strany: 2009, 299 - 306

Given any positive integer $n$, we prove the existence of infinitely many right triangles with area $n$ and side lengths in certain number fields. This generalizes the famous congruent number problem. The proof allows the explicit construction of these triangles; for this purpose we find for any positive integer $n$ an explicit cubic number field $\Bbb Q(λ)$ (depending on $n$) and an explicit point $P_λ$ of infinite order in the Mordell-Weil group of the elliptic curve $Y²=X³-n²X$ over $\Bbb Q(λ)$.

ISO 690:

Girondo, E., González-Diez, G., González-Jiménez, E., Steuding, R., Steuding, J. 2009. Right triangles with algebraic sides and elliptic curves over number fields. In Mathematica Slovaca, vol. 59, no.3, pp. 299-306. 0139-9918. DOI: https://doi.org/10.2478/s12175-009-0126-3

APA:

Girondo, E., González-Diez, G., González-Jiménez, E., Steuding, R., Steuding, J. (2009). Right triangles with algebraic sides and elliptic curves over number fields. Mathematica Slovaca, 59(3), 299-306. 0139-9918. DOI: https://doi.org/10.2478/s12175-009-0126-3