相关论文: Pythagorean Triples and a New Pythagorean Theorem
We prove that there are infinitely many integral right triangle and $\theta$-integral rhombus pairs with a common area and a common perimeter by the theory of elliptic curves.
It is well known that Pythagorean triples can be parametrized by two triples of polynomials with integer coefficients. We show that no single triple of polynomials with integer coefficients in any number of variables is sufficient, but that…
We study equivalence relation of the set of triangles generated by similarity and operation on a triangle to get a new one by joining division points of three edges with the same ratio. Using the moduli space of similarity classes of…
The aim of this paper is to introduce a new family of elliptic curves in the form of $y^2=x(x-a^2)(x-b^2)$ that have positive ranks. We first generate a list of pythagorean triples $(a,b,c)$ and then construct this family of elliptic…
We evaluate in closed form several classes of finite trigonometric sums. Two general methods are used. The first is new and involves sums of roots of unity. The second uses contour integration and extends a previous method used by two of…
In this paper we give a description of all Pythagorean triples in the ring ${{\mathbb Z}}[\tau]$. We also consider triples in the Fibonacci model set which satisfy the Diophantine equations arising from Fermat's Last Theorem. Examples are…
We introduce the Primary Gasing Triangle, a right triangle with a hypotenuse of 1 unit, to define the primary trigonometric functions: sine and cosine. This triangle serves as the foundational element in a new approach to learning…
Two well-studied Diophantine equations are those of Pythagorean triples and elliptic curves; for the first, we have a parametrization through rational points on the unit circle, and for the second we have a structure theorem for the group…
In this paper we use an elementary approach by using numerical semigroups (specifically, those with two generators) to give a formula for the number of integral points inside a right-angled triangle with rational vertices. This is the basic…
We study the triangle inequalities for angles (with different definitions) and present inequalities concerning the entries of correlation matrices through the positivity of $3\times 3$ matrices. We extend our discussions to the inequalities…
We establish some new theorems on pentagon and pentagram.
In any triangle, the perpendicular side bisectors meet the corresponding internal angle bisectors on the circumcircle. If we take those three points as the vertices of a new triangle and repeat the operation indefinitly, we end up in the…
We construct a unilateral lattice tiling of $\mathbb{R}^n$ into hypercubes of two differnet side lengths $p$ or $q$. This generalizes the Pythagorean tiling in $\mathbb{R}^2$. We also show that this tiling is unique up to symmetries, which…
A rational spherical triangle is a triangle on the unit sphere such that the lengths of its three sides and its area are rational multiples of $\pi$. Little and Coxeter have given examples of rational spherical triangles in 1980s. In this…
It is well known that Heron's theorem provides an explicit formula for the area of a triangle, as a symmetric function of the lengths of its sides. It has been extended by Brahmagupta to quadrilaterals inscribed in a circle (cyclic…
The traditional construction of primitive Pythagorean triples by the formulas of two independent variables does not allow their ordering. The paper shows a new view on the construction of primitive Pythagorean triples. A method for…
We establish a Pythagorean theorem for the absolute values of the blocks of a partitioned matrix. This leads to a series of remarkable operator inequalities.
Let a polygon be composed of equal rectangles. We find all quadratic irrationals r for which the polygon can be tiled by similar rectangles with given side ratio r.
New fundamental mathematical structures are introduced by the triples (left semistructure,right semistructure,bisemistructure) associated with the classical mathematical structures and such that the bisemistructures,resulting from the…
This note contains a short proof of a classical result: any rational symplectic matrix can be put in diagonal form after right and left multiplication by integral symplectic matrices.