Kodai Mathematical Journal

On θ-congruent numbers on real quadratic number fields

Ali S. Janfada and Sajad Salami

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Let K = Q ( $\sqrt{m}$) be a real quadratic number field, where m > 1 is a squarefree integer. Suppose that 0 < θ < π has rational cosine, say cos(θ) = s/r with 0 < | s| < r and gcd( r,s) = 1. A positive integer n is called a ( K,θ)-congruent number if there is a triangle, called the ( K,θ, n)-triangles, with sides in K having θ as an angle and nα θ as area, where α θ = $\sqrt{r^2-s^2}$. Consider the ( K,θ)-congruent number elliptic curve E n: y 2 = x( x + ( r + s) n) ( x − ( rs) n) defined over K. Denote the squarefree part of positive integer t by sqf( t). In this work, it is proved that if m ≠ sqf(2 r( rs)) and mn ≠ 2, 3, 6, then n is a ( K,θ)-congruent number if and only if the Mordell-Weil group E n( K) has positive rank, and all of the ( K, θ, n)-triangles are classified in four types.

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Kodai Math. J., Volume 38, Number 2 (2015), 352-364.

First available in Project Euclid: 9 July 2015

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Janfada, Ali S.; Salami, Sajad. On θ-congruent numbers on real quadratic number fields. Kodai Math. J. 38 (2015), no. 2, 352--364. doi:10.2996/kmj/1436403896. https://projecteuclid.org/euclid.kmj/1436403896

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