Rational Right Triangles of a Given Area

Abstract

Starting from any given rational-sided, right triangle, for example, the (3,4,5)-triangle with area 6, we use Euclidean geometry to show that there are infinitely many other rational-sided, right triangles of the same area. We show further that the set of all such triangles of a given area is finitely generated under our geometric construction. Such areas are known as “congruent numbers” and have a rich history in which all the results in this article have been proved and far more. Yet, as far as we can tell, this seems to be the first exploration using this kind of geometric technique.

Keywords:

• MSC: Primary 11A99
• Secondary 11G05

Acknowledgments

The author thanks her Ph.D. advisor, Andrew Granville, for his guidance and inspiring discussions in the development of this article. She is also grateful to Jennifer Balakrishnan, John Coates, and Henri Darmon for their helpful comments and suggestions. The author was supported by the European Research Council grant agreement No. 670239.

Additional information

Notes on contributors

Stephanie Chan

STEPHANIE CHAN completed her master’s degree at the University of Oxford in 2016. She is currently a Ph.D. student at University College London. Her research interests mainly lie in number theory.