Heron triangles and van Luijks Theorem

Abstract

A Heron triangle is a triangle with integer sides and integer area. A rational triangle is a triangle with rational sides and area. We will set out to prove van Luijks Theorem, that there exist infinitely many non-similar rational triangles with the same area and perimeter. Unlike the original proof, we will not use elliptic surfaces, using instead only elliptic curves. Hopefully making the proof a bit more accessible to readers without a background in algebraic geometry. We will also prove, as a corollary, that there exist arbitrarily many Heron triangles having the same area and perimeter, and construct a method for generating such triangles. This thesis will also provide a fix to a minor flaw in the original proof of the theorem.