Right triangles with algebraic sides and elliptic curves over number fields

Details Right triangles with algebraic sides and elliptic curves over number fields Right triangles with algebraic sides and elliptic curves over number fields

2009-03-26

arxiv.org/abs/0903.4611v1

Math. Slovaca 59, no. 3, 299-306 (2009)

Mathematics

Number Theory

To appear in Math. Slovaca

Scientific paper

10.2478/s12175-009-0126-3

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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