A Unique Pair of Triangles —Proof of a Simple Theorem Using Abstract Modern Mathematics—
October 15, 2018
Yoshinosuke Hirakawa and Hideki Matsumura, graduate students of the Keio Institute of Pure and Applied Sciences (KiPAS) Arithmetic Geometry and Number Theory Group at the Keio University Faculty of Science and Technology, have proven a new theorem that states there is only one pair (up to similitude) of an isosceles triangle and a right triangle for which the lengths of all its sides are integers and which have the same perimeter and the same area.
The lengths of lines and areas of figures are basic geometrical quantities that are indispensable when measuring everything around us. For example, a familiar figure from textbooks is a right triangle with sides of lengths 3, 4, and 5 respectively. Moreover, an important question that has been studied since the time of ancient Greece is how many right triangles there are for which the lengths of all sides are integers. One field of modern mathematics which has greatly developed in the twentieth century under the influence of this tradition is arithmetic geometry.
The above theorem was proved by applying the "theory of p-adic Abelian integrals" and "descent of rational points" in arithmetic geometry. It is rare that highly abstract modern mathematics has applications to such familiar objects.
The above research is to be published under the title of "A unique pair of triangles" in the "Journal of Number Theory." (An electronic version has already been released as an “article in press” on August 24, 2018.)