Eugen J. Ionascu, Florian Luca, Pantelimon Stanica
In this paper, we study the function $H(a,b)$, which associates to every pair of positive integers $a$ and $b$ the number of positive integers $c$ such that the triangle of sides $a,b$ and $c$ is Heron, i.e., has integral area. In particular, we prove that $H(p,q)\le 5$ if $p$ and $q$ are primes, and that $H(a,b)=0$ for a random choice of positive integers $a$ and $b$.
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An Infinite Family of Primitive Heron Triangles with Two Sides as Perfect Squares
该研究通过椭圆曲线理论证明存在无限多个原始海伦三角形，其中两边为完全平方数，并暗示可能存在三边均为完全平方数的原始海伦三角形。
Given any positive integer n, it is well known that there always exist triangles with rational sides a, b and c such that the area of the triangle is n. Assuming finiteness of the Shafarevich-Tate group, we first construct a family of infinitely many Heronian elliptic curves of rank exactly 1 from Heron triangles of a certain type. We also explicitly produce a separate family of infinitely many Heronian elliptic curves with 2-Selmer rank lying between 1 and 3.
We describe several algorithms for the generation of integer Heronian triangles with diameter at most $n$. Two of them have running time $\mathcal{O}\left(n^{2+\varepsilon}\right)$. We enumerate all integer Heronian triangles for $n\le 600000$ and apply the complete list on some related problems.
By Fermat's method, we show that there are infinitely many Heron triangle and $θ$-integral rhombus pairs with a common area and a common perimeter. Moreover, we prove that there does not exist any integral isosceles triangle and $θ$-integral rhombus pairs with a common area and a common perimeter.
The aim of this paper is to introduce a new family of elliptic curves with positive ranks. These elliptic curves have been constructed with certain rational numbers, namely a, b, and c as sides of Heron triangles having rational areas $k$. It turned out that the torsion groups of this family are of the form $\frac{\Bbb{Z}}{2\Bbb{Z}}\times \frac{\Bbb{Z}}{2\Bbb{Z}}$ and also the rank is positive.
Given any positive integer $n$, it is well-known that there always exists a triangle with rational sides $a,b$ and $c$ such that the area of the triangle is $n$. For a given prime $p \not \equiv 1$ modulo $8$ such that $p^{2}+1=2q$ for a prime $q$, we look into the possibility of the existence of the triangles with rational sides with $p$ as the area and $\frac{1}{p}$ as $\tan \fracθ{2}$ for one of the angles $θ$. We also discuss the relation of such triangles with the solutions of certain Diophantine equations.
In "Unsolved Problems in Number Theory" problem D22 Richard Guy asked for the existence of simplices with integer lengths, areas, volumes... In dimension two this is well known, these triangles are called Heron triangles. Here I will present my results on Heron tetrahedra, their connection to the existence of an integer box (problem D18), the tools for the search for higher dimensional Heron simplices and my nice embedding conjecture about Heron simplices, which I can only proof in dimension two, but I verified it for a large range in dimension three.
A generalization of the congruent number problem is to find positive integers $n$ that appear as the areas of Heron triangles. Selmer group of a congruent number elliptic curve has been studied quite extensively. Here, we look into the $2$-Selmer group structure for Heronian elliptic curves associated with Heron triangles of area $n$ and one of the angle $θ$ such that $\tan \fracθ{2} = n^{-1}$ and $n^{2}+1=2q$ for some prime $q$.
We define spherical Heron triangles (spherical triangles with "rational" side-lengths and angles) and parametrize them via rational points of certain families of elliptic curves. We show that the congruent number problem has infinitely many solutions for most areas in the spherical setting and we find a spherical Heron triangle with rational medians. We also explore the question of spherical triangles with a single rational median or a single a rational area bisector (median splitting the triangle in half), and discuss various problems involving isosceles spherical triangles.
We define hyperbolic Heron triangles (hyperbolic triangles with "rational" side-lengths and area) and parametrize them in two ways as rational points of certain elliptic curves. We show that there are infinitely many hyperbolic Heron triangles with one angle $α$ and area $A$ for any (admissible) choice of $α$ and $A$; in particular, the congruent number problem has always infinitely many solutions in the hyperbolic setting. We also explore the question of hyperbolic triangles with a rational median and a rational area bisector (median splitting the triangle in half).
A Heron quadrilateral is a cyclic quadrilateral whose area and side lengths are rational. In this work, we establish a correspondence between Heron quadrilaterals and a family of elliptic curves of the form $y^2 = x3+/alpha x^2-n^2x.$ This correspondence generalizes the notions of Goins and Maddox who established a similar connection between Heron triangles and elliptic curves. We further study this family of elliptic curves, looking at their torsion groups and ranks. We also explore their connection with congruent numbers, which are the /alpha = 0 case. Congruent numbers are positive integers…
Triangles with integer length sides and integer area are known as Heron triangles. Taking rescaling freedom into account, one can apply the same name when all sides and the area are rational numbers. A perfect triangle is a Heron triangle with all three medians being rational, and it is a longstanding conjecture that no such triangle exists. However, Buchholz and Rathbun showed that there are infinitely many Heron triangles with two rational medians, an infinite subset of which are associated with rational points on an elliptic curve $E(\mathbb{Q})$ with Mordell-Weil group $\mathbb{Z}\times \m…
Heron triangles and the hunt for unicorns
针对完美三角形（三条中线均有理的Heron三角形）不存在的猜想，研究给出了具有两条有理中线的近乎完美Heron三角形的显式公式，该公式由Somos序列表示，且这类三角形与椭圆曲线E(Q)上的有理点相关。
This article highlights interactions of diverse areas: the Heron formula for the area of a triangle, the Descartes circle equation, and right triangles with integer or rational sides. New and old results are synthesized. We show that every primitive Pythagorean triple (PPT), furnishes a Descartes quadruple of tangent circles with integral curvatures, and so generates an integral Apollonian packing (IAP) containing a rectangle of centers. Thus Pythagorean triples serve to generate a large number of in-equivalent integral packings.
The Arithmetic Geometry of Square-Sided Heron Triangles
本文研究了具有三个完全平方边的海伦三角形的等价条件，并通过代数曲线的有理点来确定其有限性，同时给出了具有两个完全平方边的海伦三角形的完整刻画及参数解族，证明了存在无限多个原始海伦三角形。
A Diophantine Criterion for the Shafarevich-Tate Groups of Elliptic Curves from Heron Triangles
通过研究丢番图方程 p u^2 = v^2 + w^2 的整数解，证明了对于特定形式的素数 p，椭圆曲线 E_p 的秩和沙法列维奇-塔特群呈现二分法。
A rational triangle is a triangle with rational sides and rational area. A Heron triangle is a triangle with integral sides and integral area. In this article we will show that there exist infinitely many rational parametrizations, in terms of s, of rational triangles with perimeter 2s(s+1) and area s(s^2-1). As a corollary, there exist arbitrarily many Heron triangles with all the same area and the same perimeter. The proof uses an elliptic K3 surface Y. Its Picard number is computed to be 18 after we prove that the Neron-Severi group of Y injects naturally into the Neron-Severi group of the …
Some Remarks on $τ$-Congruent Numbers
本文扩展了Chahal的工作，确定了紧外切单位圆的所有Heron三角形及其生成的$\tau$-同余数，刻画了紧外切单位椭圆的所有有理直角三角形并识别了相应的同余数，还研究了有理直角三角形顶点对应的旁切圆产生的同余数。
Following in Yiu's Footsteps but on the Eisenstein Lattice
本文将 Yiu 关于海伦三角形可在整数格上实现的结论推广到爱森斯坦格，证明了顶点在爱森斯坦格上的三角形具有相应的可实现性结果。
We study some Diophantine problems related to triangles with two given integral sides. We solve two problems posed by Zoltán Bertalan and we also provide some generalization.
In this paper we consider the problem of finding pairs of triangles whose sides are perfect squares of integers, and which have a common perimeter and common area. We find two such pairs of triangles, and prove that there exist infinitely many pairs of triangles with the specified properties.
We have computed a table of the triangle sides of all congruent numbers less than 10,000, which improves and extends the existing public table. We give some background on properties of the triangle sides, and explain how we computed our table, which is available on https://github.com/dgpaloalto/Congruent-Numbers
It is well known that a triangle with side lengths 3, 4 and 5 is right-angled. Euclid was the first to give a formula for generating other right-angled triangles with integer side lengths. In this text, I present a novel algorithm to generate all possible right-angled triangles with integer side lengths, in which the three side lengths have no common divisor. The algorithm is based on the difference in length between the hypothenuse and the largest of the two other sides. I also prove the completeness of this algorithm: it generates all possible such triangles and nothing but such triangles.
For a triangle $Δ$, let (P) denote the problem of the existence of points in the plane of $Δ$, that are at rational distance to the 3 vertices of $Δ$. Answer to (P) is known to be positive in the following situation: $Δ$ has one rational side and the square of all sides are rational. Further, the set of solution-points is dense in the plane of $Δ$. See [3] The reader can convince himself that the rationality of one side is a reasonable minimum condition to set out, otherwise problem (P) would stay somewhat hazy and scattered. Now, even with the assumption of one rational side, problem (P) stay…
Given any positive integer n, we prove the existence of infinitely many right triangles with area n and side lengths in certain number fields. This generalizes the famous congruent number problem. The proof allows the explicit construction of these triangles; for this purpose we find for any positive integer n an explicit cubic number field Q(λ) (depending on n) and an explicit point P_λof infinite order in the Mordell-Weil group of the elliptic curve Y^2=X^3-n^2*X over Q(λ).
We study the existence of equilateral triangles of given side lengths and with integer coordinates in dimension three. We show that such a triangle exists if and only if their side lengths are of the form $\sqrt{2(m^2-mn+n^2)}$ for some integers $m,n$. We also show a similar characterization for the sides of a regular tetrahedron in $\Z^3$: such a tetrahedron exists if and only if the sides are of the form $k\sqrt{2}$, for some $k\in\N$. The classification of all the equilateral triangles in $\Z^3$ contained in a given plane is studied and the beginning analysis is presented. A more general pa…
A rational triangle is a triangle with rational side lengths. We consider three different families of rational triangles having a fixed side and whose vertices are rational points in the plane. We display a one-to-one correspondence between each family and the set of rational points of an algebraic curve. These algebraic curves are: a curve of genus 0, an elliptic curve, and a genus 3 curve. We study the set of rational points on each of these curves and describe some of its rational points explicitly.
A perfect triangle is a triangle with rational sides, medians, and area. In this article, we use a similar strategy due to Pocklington to show that if $Δ$ is a perfect triangle, then it cannot be an isosceles triangle. It gives a partial answer to a question of Richard Guy, who asked whether any perfect triangles exist. No example has been found to date. It is widely believed that such a triangle does not exist.
A triangle with rational sides and rational area is called a rational triangle. In this paper we consider three problems of finding pairs of rational triangles which have a common circumradius as well as either a common perimeter or a common inradius or a common area. While several similar problems concerning pairs of rational triangles have been considered in the existing literature, these three problems have not been studied till now. For each of these problems, we give a parametric solution and we also indicate how additional parametric solutions of these problems may be obtained.
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.
Rational Angle Bisection Problem in Higher Dimensional Spaces and Incenters of Simplices over Fields
本文将有理角平分问题推广到n维空间k^n中，研究了在k子域下，两直线方向向量的角平分线方向向量的条件，并推导了积分解公式，讨论了单纯形内心为k有理数的充要条件及三角形性质。
We consider the problem of finding integer-sided triangles with R/r an integer, where R and r are the radii of the circumcircle and incircle respectively. We show that such triangles are relatively rare.
It is easy to find a right-angled triangle with integer sides whose area is 6. There is no such triangle with area 5, but there is one with rational sides (a `\emph{Pythagorean triangle}'). For historical reasons, integers such as 6 or 5 that are (the squarefree part of) the area of some Pythagorean triangle are called `\emph{congruent numbers}'. These numbers actually are interesting for the following reason: Notice the sequence $\frac14$, $6\frac14$, $12\frac14$. It is an arithmetic progression with common difference 6, consisting of squares $(\frac12)^2$, $(\frac52)^2$, $(\frac72)^2$ of rat…
Triangles with one fixed side-length, a Furstenberg type problem, and incidences in finite vector spaces
本文利用有限域中点与刚体运动的关联界，建立固定边长三角形及高维单形全等类的尖锐存在条件，并统一证明已知最佳单形分布结果，同时研究刚体运动下的 Furstenberg 型问题。
We confirm the list from \cite{MSS} of values $D$ for which the high-density hard-core model on $\mathbb{Z}^2$ with exceptional distance $D$ has infinitely many extremal Gibbs states. As a byproduct, we prove that for all $D>0$ there exists an acute-angled triangle inscribes in $\mathbb{Z}^2$ with side-lengths at least $D$ and area $\sqrt{3}/4\cdot D^2+O(D^{4/5})$.
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