A rational triangle is a triangle with sides of rational lengths. In this short note, we prove that there exists a unique pair of a rational right triangle and a rational isosceles triangle which have the same perimeter and the same area. In the proof, we determine the set of rational points on a certain hyperelliptic curve by a standard but sophisticated argument which is based on the 2-descent on its Jacobian variety and Coleman's theory of p-adic abelian integrals.

In this paper, we construct some families of infinitely many hyperelliptic curves of genus 2 with exactly two rational points. In the proof, we first show that the Mordell-Weil ranks of these hyperelliptic curves are 0 and then determine the sets of rational points by using the Lutz-Nagell type theorem for hyperelliptic curves which was proven by Grant.Contents2), p ≡ −3 (mod 8) 18 3. Application of the Lutz-Nagell type theorem for hyperelliptic curves 21 4. Concluding remarks 23 Appendix A. 23 References 24

No abstract

No abstract

No abstract