September 12, 2018
Keio University
Yoshiyuki Hirakawa (a third-year student in the Doctoral Programs) and Hideki Matsumura (a second-year student in the Doctoral Programs) of the KiPAS Number Theory and Geometry Group at the Keio University Graduate School of Science and Technology have successfully proven a previously unknown theorem: "among pairs of a right-angled triangle and an isosceles triangle where all side lengths are integers, there is only one pair (up to similarity) that has both the same perimeter and the same area."
The length of lines and the area of figures are fundamental "geometric" objects essential for measuring things in our daily lives. For example, a right-angled triangle with side lengths of 3, 4, and 5 is a familiar figure from textbooks, but the question of how many right-angled triangles have all "integer" side lengths was an important problem studied in ancient Greece. "Number theory and geometry" is a field of modern mathematics that developed significantly in the 20th century, following this line of inquiry.
In this study, we succeeded in proving the theorem mentioned at the beginning by applying "p-adic Abel's theorem" and the "method of descent for rational points" from number theory and geometry. It is extremely rare to find such a familiar application in highly abstract modern mathematics, making this a valuable research achievement.
The results of this research are set to be published in the American academic journal specializing in number theory, "Journal of Number Theory," as the paper "A unique pair of triangles" (the electronic version was already published as an article in press on August 24, 2018).
Please see below for the full press release.