2025/12/17 by Halbeisen, Lorenz, Hungerbühler, Norbert, Zargar, Arman Shamsi
Mathematics · Computer Science · #Mathematics and Applications #Algebraic Geometry and Number Theory #Polynomial and algebraic computation
paper · doi:10.48550/arxiv.2512.15237
We consider the problem of finding integer triangles with R/r a positive rational, where R and r are the radii of the circumcircle and an excircle, respectively. We show that for general triangles R/r>1/4 applies. The equation R/r=N turns out to be related to the elliptic curve EN given by v2=u3+2(2N2+2N-1)u2-(4N-1)u. If N>1/4 is rational, then the torsion group of EN is \mathbb Z/2\mathbb Z×\mathbb Z/6\mathbb Z if N(N+2) is a square and \mathbb Z/6\mathbb Z otherwise. We show that a rational triangle with rational ratio R/r=N exists if and only if N>1/4 and there exists a rational non-torsion point on the curve EN which satisfies a certain condition. Furthermore, we show that the rank of EN is positive when N = m2 ± 1>1/4 for a rational m. We also show that on every curve EN whose rank is positive, there are infinitely many rational points which lead to infinitely many non-similar integer triangles with R/r=N.