H. Lee Price, Frank R. Bernhart
Given a right triangle and two inscribed squares, we show that the reciprocals of the hypotenuse and the sides of the squares satisfy an interesting Pythagorean equality. This gives new ways to obtain rational(integer)right triangles from a given one.
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A note on Pythagorean Triples
本文通过直角边的基本刻画，建立了毕达哥拉斯三元组之间的关系，特别是与相同直角边升至整数幂时的三元组的关系。
It is well-known that pythagorean triples can be represented by points of the unit circle with rational coordinates. These points form an abelian group, and we describe its structure. This structural description yields, almost immediately, an enumeration of the normalized pythagorean triples with a given hypotenuse, and also to an effective method for producing all such triples. This effective method seems to be new. This paper is intended for the general mathematical audience, including undergraduate mathematics students, and therefore it contains plenty of background material, some history a…
In this paper we introduce a formula that parameterises the Pythagorean triples as elements of two series. With respect to the standard Euclidean formula, this parameterisation does not generate the Pythagorean triples where the elements of the triple are all divisible by 2. A necessary and sufficient condition is also proposed for a Pythagorean triple obtained from this formula to be primitive.
A review of Pythagorean triples from both classical and modern viewpoints
本文从经典与现代视角综述了勾股数的研究方法，讨论了非交换与算子理论方法，并提出了未来研究方向。
New formulas and trees of Pythagorean triples
本文提出了构造勾股数的新公式，将其推广至高次幂方程，并应用于n=4的Fermat方程等问题。
Ever since it was published by Neugebauer and Sachs in 1945, the Old Babylonian tablet known as Plimpton 322 has been the subject of numerous studies leading to different and often conflicting interpretations of it. Overall, the tablet is more or less viewed as a list of fifteen Pythagorean triplets, but scholars are divided on how and why the list was devised. In this paper, we present a survey of previous attempts to interpret Plimpton 322, and then offer some new insights that could help in sharpening the endless debate about this ancient tablet.
This paper summarizes basic properties of PPTs and shows that each PPT belongs to one of six different classes. Mapping an ordered sequence of PPTs into a corresponding sequence of these six classes makes it possible to use them in cryptography. We pose problems whose solution would facilitate such cryptographic application.
Matrices with integer eigenvalues for all permutations of coefficients (thanks to Pythagoras!)
利用毕达哥拉斯三元组生成具有整数特征值的所有系数排列的矩阵，通过简单公式实现
We motivate and then prove a generalized pythagorean theorem for parallelepipeds in Euclidean space.
In this note we show that in addition to two integers forming a Pythagorean triple, there also exist two irrational numbers in terms of which this Pythagorean triple can also be obtained. We also put forward a relation between these two pairs and the hypotenuse of a Pythagorean triangle.
In this article using elementary school level Geometry we observe an alternative proof of Pythagorean Theorem from Heron's Formula.
Pythagorean Theorem in Elamite Mathematics
本文考察毕达哥拉斯定理在苏萨数学文本中的应用，重点分析SMT No.1中的几何应用实例。
Pythagorean Theorem, Law of Sines and Law of Cosines: alternative proofs via shape derivatives
本文提出利用形状导数，通过沿特定方向平移三角形并令面积变化为零，统一证明勾股定理、正弦定理和余弦定理。
Trigonometric Ratios Can Prove the Pythagorean Theorem
受 Jackson 和 Johnson 启发，本文基于三角恒等式给出毕达哥拉斯定理的三个非循环证明，并指出二倍角公式的关键作用。
Dissecting Circles to Prove a Square: A Novel Geometric Proof of the Pythagorean Theorem Using Circular Segments and Area Decomposition
本文提出一种毕达哥拉斯定理的新几何证明，通过三个外接圆、圆弓形面积分解及三角恒等式，以两种方式计算斜边半圆面积，且该方法未见于既有文献。
A New Transformation of the Magic Square of Squares
本文证明高斯平方算术三元组与毕达哥拉斯三元组呈3对1对应，借此将平方幻方难题解转换为更大的完美高斯平方结构，并反向生成高斯整数中的非平凡近似解。
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…
The Pythagorean triples have the structure of a ternary rooted tree; the tree is based on the Cayley graph of a free subgroup of the modular group
We show that Leonardo da Vinci's well known proof of the Pythagorean theorem is due to Mayer and not to da Vinci.
In his treatise addressed to Dositheus of Pelusium, Archimedes of Syracuse obtained the result of which he was the most proud: a sphere has two-thirds the volume of its circumscribing cylinder. At his request a sculpted sphere and cylinder were placed on his tomb near Syracuse. Usually, it is admitted that to find this formula, Archimedes used a half polygon inscribed in a semicircle; then he performed rotations of these two figures to obtain a set of trunks in a sphere. This set of trunks allowed him to determine the volume. In our opinion, Archimedes was so clever that he found the proof wit…
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