分野別セミナー

A positive integer $n$ is called a $\theta$-congruent number if there is triangle with rational sides $a,b$ and $c$ for which the angle between $a$ and $b$ is equal to $\theta$ and its area is $n\sqrt{r^2 - s^2}$, where $0 < \theta < \pi$, $\cos \theta = s/r$ and $0 \leq |s| < r$ are relatively prime integers. The notion of $\theta$-congruent numbers is a natural generalization of the classical congruent numbers, which correspond to the case where $\theta = \pi/2$. It is known that the problem of classifying $\theta$-congruent numbers is related to the problem of finding non-trivial rational points on certain families of elliptic curves. In this talk, we present a certain variant of the congruent number problem. More explicitly, we discuss integers which occur as areas of triangles with two rational sides and arbitrary fixed angle $\psi$ with one adjacent side a rational multiple of a quadratic surd. We call such numbers $\psi$-congruent. We present a criterion that involves elliptic curves for deciding whether a given positive integer is $\psi$-congruent. We also discuss some results about $\pi/4$-congruent numbers from a joint work with Soma Purkait.