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arXiv · 0903.4611

Right triangles with algebraic sides and elliptic curves over number fields

Ernesto Girondo·Gabino Gonzalez-Diez·Enrique Gonzalez-Jimenez·Rasa Steuding·Jorn Steuding
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Abstract

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 Q(\lambda) (depending on n) and an explicit point P_\lambda of infinite order in the Mordell-Weil group of the elliptic curve Y^2=X^3-n^2*X over Q(\lambda).

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Ernesto Girondo, Gabino Gonzalez-Diez, Enrique Gonzalez-Jimenez, Rasa Steuding, Jorn Steuding. 2009-03-26. Right triangles with algebraic sides and elliptic curves over number fields. https://doi.org/10.2478/s12175-009-0126-3

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