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 there exists a parametrization of Pythagorean triples by a single triple of integer-valued polynomials. The second author has recently studied polynomial pa...

We address the question of the “partition regularity” of the Pythagorean equation a+b = c; in particular, can the natural numbers be assigned a 2-coloring, so that no Pythagorean triple (i.e., a solution to the equation) is monochromatic? We prove that the hypergraph of Pythagorean triples can contain no Steiner triple systems, a natural obstruction to 2-colorability. Then, after transforming t...

Does the world really need another article about Pythagorean triples? Here is why we think so. The set of Pythagorean triples has a lot of interesting structure, which has intrigued both amateur and professional mathematicians. It is the topic of an extensive mathematical literature, almost all of which relies on an enumeration of primitive Pythagorean triples that has been known since ancient ...

The purpose of this study was to determine the mathematical representation ability eighth grade junior high school students on Pythagorean theorem material based 3 indicators, namely visual representation, symbolic and verbal representation. This used subject class VIII D at SMP 1 Mejobo. research method uses descriptive with a qualitative approach. instrument is test Theorem material. data ana...

This short note describes a development of the traditional Pythagorean tree fractal to produce interesting three-dimensional structures based on cubes rather than squares. Adding a 90˚rotation at each bifurcation encourages three-dimensional growth, creating increasingly artistic shapes as the branching angle decreases. 1. Pythagorean box trees The Pythagorean tree is a plane fractal constructe...

A Pythagorean triple is a triple of positive integers a,b,c ∈ N+ satisfying a2 + b2 = c2. Is it true that, for any finite coloring of N+, at least one Pythagorean triple must be monochromatic? In other words, is the Diophantine equation X2 +Y 2 = Z2 regular? This problem has been open since several decades, even restricted to 2-colorings. In this note, we introduce partial morphisms, which are ...

Minkowski Pythagorean hodograph curves are polynomial curves with polynomial speed, measured with respect to Minkowski norm. Curves of this special class are particularly well suited for representing medial axis transforms of planar domains. In the present paper we generalize this polynomial class to a rational class of curves in Minkowski 3-space. We show that any rational Minkowski Pythagorea...

A lemniscate is a curve defined by two foci, F1 and F2. If the distance between focal points of - F2 2a (a: constant), then any point P on satisfy equation PF1 PF2 = a^2. Jacob Bernoulli first described in 1694. The Fagnano discovered double angle formula lemniscate(1718). Euler extended Fagnano's to more general addition theorem(1751). function was subsequently proposed Gauss around year 1800....