Heron triangles and their elliptic curves

doi:10.1016/j.jnt.2019.12.005

Journal of Number Theory

Volume 213, August 2020, Pages 232-253

https://doi.org/10.1016/j.jnt.2019.12.005 Get rights and content

Abstract

In geometry, a Heron triangle is a triangle with rational side lengths and integral area. We investigate Heron triangles and their elliptic curves. In particular, we provide some new results concerning Heron triangles and give elementary proofs for some results concerning Heronian elliptic curves.

Introduction

Motivated by Goins [2] and based on our research [4], [5], we investigate Heron triangles and the corresponding elliptic curves. In particular, we give elementary proofs for some results in Goins and Maddox [3] and generalize the main results of [4] and [5].

In Section 2, we introduce and investigate H-triples: A rational triple $(a, b, λ) \in Q^{3}$ is called an H-triple if $a, b$ are non-zero and $c : = \sqrt{a^{2} - 2 λ a b + b^{2}}$ is rational. This notion generalizes Heron triples, for which $\sqrt{1 - λ^{2}}$ is a positive rational number, and for which a Heron triangle exists, i.e., a triangle with integral area and rational sides $| a |, | b |, c$ such that the cosine of the angle opposite of side c is λ. We start by giving a relation between solutions of $(p^{2} - q^{2}) (a^{2} - b^{2}) = r^{2} - s^{2}$ in positive integers and the existence of certain pairs of H-triples. In particular, in Proposition 2 we show that non-zero integers $p, q, a, b, r, s$ are an integral solution for the Diophantine equation $(p^{2} - q^{2}) (a^{2} - b^{2}) = r^{2} - s^{2}$ if and only if there is a rational $λ \in Q$ such that both, $(q a, p b, λ)$ and $(p a, q b, λ)$ , are H-triples.

Based on Section 2 we investigate families of Heron triangles sharing a given angle in Section 3. In particular, we first formulate an algorithm which generalizes a result of Fermat's, to produce infinitely many Heron triangles sharing the same angle and the same area. Then, we characterize isosceles Heron triangles by showing in Theorem 6 that there is an isosceles Heron triangle $(a, b, c)$ with $a = b$ if and only if there are $u, v \in N$ such that $\frac{v^{2} - u^{2}}{u^{2} + v^{2}} = \frac{a^{2} + b^{2} - c^{2}}{2 a b}$ and $u^{2} + v^{2}$ is a square. Furthermore, we investigate pairs of integral isosceles Heron triangles and integral Pythagorean triangles of the same area, and show in Proposition 7 that every positive integral solution of the Diophantine equation $p q (p^{2} - q^{2}) = 2 m n (m^{2} - n^{2}) \cdot □$ (where □ denotes a square) leads to such a pair. Non-trivial solutions of this Diophantine equation are given in Corollary 8. Finally, by generalizing a result from [5], we construct triples of integral Heron triangles of the same area and sharing an angle from positive solutions of the Diophantine equation $m = n^{2} + n l + l^{2}$ .

In Section 4, we investigate the torsion group and the rank of elliptic curves related to Heron triangles, so-called Heronian elliptic curves, which are curves of the form $E_{u, v, A} : y^{2} = x^{3} + \frac{v^{2} - u^{2}}{u v} A x^{2} - A^{2} x, where A is a positive integer.$ Here we use results from Section 3. In particular, we provide a new proof for Theorem 15 which states that the torsion group of a Heronian elliptic curve $E_{u, v, A}$ is isomorphic either to $Z / 2 Z \times Z / 2 Z$ or — in the case when there exists an isosceles Heron triangle with area A — to $Z / 2 Z \times Z / 4 Z$ . The proof given by Goins and Maddox [3, Proposition 3.3] relies on Mazur's Theorem, which states that the torsion group of an elliptic curve is isomorphic to one of fifteen groups, and uses twice a symbolic computer package (e.g., MAGMA) in order to show that Heronian elliptic curves never have rational points of order 3 or of order 8. However, in the proof given below, we do not need computer assistance, and in the case when u and v are both odd, we do not even use Mazur's Theorem. At the end of this article, we show — by generalizing a result from [5] — under which conditions positive solutions of $m = n^{2} + n l + l^{2}$ lead to Heronian elliptic curves of rank at least 2 and provide examples of Heronian elliptic curves of rank 5.

Section snippets

A theorem of Sós

A rational triple $(a, b, λ) \in Q^{3}$ , where $a, b$ are non-zero, is called a H-triple if $c : = \sqrt{a^{2} - 2 λ a b + b^{2}} is rational.$

Theorem 1 Sós

For every H-triple $(a, b, λ)$ there are relatively prime integers $m, n \in Z$ and a rational $μ \in Q$ , such that $a = μ (m^{2} - n^{2}) b = μ (2 m (n + λ m)) c = μ (m^{2} + 2 λ m n + n^{2})$

For a proof see Sós [11, p. 189].

Notice that for $λ = 0$ the set of equations (1) corresponds to the well-known formula for rational Pythagorean triples. Notice also that for $- 1 < λ < 1$ , the values $| a |, | b |, | c |$ are the side lengths of a triangle with $\cos (θ) = λ$ , where

Heron triples

An H-triple $(a, b, λ) \in Q^{3}$ is called a Heron triple if $\bar{λ} : = \sqrt{1 - λ^{2}} \in Q^{+} and A : = \bar{λ} \frac{| a b |}{2} \in N .$

Notice that if $(a, b, λ)$ is a Heron triple, then $λ = \frac{1 - τ^{2}}{1 + τ^{2}}$ for some $τ \in Q$ . If we set $τ = \frac{u}{v}$ , where we always assume that u and v are relatively prime, then $λ = \frac{v^{2} - u^{2}}{u^{2} + v^{2}} and \bar{λ} = \frac{2 u v}{u^{2} + v^{2}} .$ In particular we obtain $A = \frac{u v}{u^{2} + v^{2}} a b,$ and since $(u, v) = 1$ and A is integral, we have that $u^{2} + v^{2}$ divides ab. In the sequel we will write $Q : = a b$ . In particular, it follows that λQ is integral.

Heron triangles

If $(a, b, λ) \in Q^{3}$ is a Heron triple and $c : = \sqrt{a^{2} - 2 λ a b + b^{2}}$ , then $| a |, |$

Heronian elliptic curves

It is well known that the rational points on an elliptic curve form an abelian group. Moreover, by Mordell's Theorem, this group is finitely generated (see, for example, Mordell [9, Ch. 16]). Therefore, by the Fundamental Theorem of Finitely Generated Abelian Groups, the group of rational points on an elliptic curve is isomorphic to some group of the form $\underset{torsion group}{\underset{︸}{Z / n_{1} Z \times \dots \times Z / n_{k} Z}} \times Z^{r},$ where $n_{1}, \dots, n_{k}$ are positive integers with $n_{i} | n_{i + 1}$ , and r is a non-negative integer. The group $Z / Z_{n_{1}} \times \dots \times Z / Z_{n_{k}}$ ,

Odds and ends

1.
Theorem 4 was just formulated for u and v both odd, which implies that $c_{n} = \frac{r}{2^{k} \cdot s}$ , where r and s are both odd and $k \geq 0$ , and includes the case when $u = v = 1$ (i.e., $λ = 0$ ). If either u or v is even, then $c_{n}$ may be of the form $2^{k} \cdot \frac{r}{s}$ where r and s are both odd and $k > 0$ . Moreover, for $u = 32$ , $v = 9$ , $a_{0} = 663$ , $b_{0} = 575$ , $c_{0} = 1192 = 2^{3} \cdot 149$ , we get $c_{1} = 2^{4} \cdot \frac{531448501}{1014541}$ , and in general, for all $n \geq 1$ , we have $c_{n} = 2^{4} \cdot \frac{r}{s}$ for some odd integers r and s.
2.
The three points on the curve $E_{λ, Q, A}$ which correspond to the three Heron

Acknowledgment

We would like to thank the referee for his or her thorough review and highly appreciate the comments and suggestions, which helped to improve the quality of the article.

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General SectionHeron triangles and their elliptic curves

Abstract

Introduction

Section snippets

A theorem of Sós

Heron triples

Heron triangles

Heronian elliptic curves

Odds and ends

Acknowledgment

J. Number Theory

J. Number Theory

Fermat's Diophanti Alex. Arith., 1670

The ubiquity of elliptic curves

Not. Am. Math. Soc.

Heron triangles via elliptic curves

Rocky Mt. J. Math.

General Section
Heron triangles and their elliptic curves