Dr. Yoshinosuke Hirakawa (3rd year) and Dr. Hideki Matsumura (2nd year) of KiPAS Mathematical Geometry Group, Graduate School of Science and Engineering, Keio University said, "Right triangles and isosceles triangles whose sides are all integers. There is only one set of right triangles (except for similarities) that has the same circumference and area. ”We succeeded in proving a previously unknown theorem.
The length of the line and the area of the figure are the basic "geometric" objects that are indispensable for surveying the things around us.For example, a right triangle with a side length of 3: 4: 5 is a familiar figure in textbooks, but the question of how many right triangles have side lengths that are all "integers" was studied in the ancient Greek era. Was an important issue."Arithmetic geometry" is a field of modern mathematics that developed significantly in the 20th century following this trend.
今回の研究では、数論幾何学における「p進Abel積分論」と「有理点の降下法」と呼ばれる手法を応用。三辺の長さの整数比が377：352：135の直角三角形と、三辺の長さの整数比が366：366：132の二等辺三角形は、比をそのまま長さとすれば、周の長さが864（＝377+352+135＝366+366+132）、面積が23760（135×352÷2＝132×360［二等辺三角形の高さ］÷2）であり同じ値になることが分かった。
It is presumed that the problem solved this time was also considered in the ancient Greek era.In the research, the "Chabauty-Coleman method" based on the p-adic Abel integral theory and the "2-descent method" were used, both of which are relatively new methods developed after the 1980s.The contrast between such simple problems and sophisticated solutions, and the results of research with a large gap between the times, are valuable achievements that enhance the beauty of modern mathematics.
