Pythagorean Triples
نویسنده
چکیده
The name comes from elementary geometry: if a right triangle has leg lengths x and y and hypotenuse length z, then x + y = z. Of course here x, y, z are positive real numbers. For most integer values of x and y, the integer x + y will not be a perfect square, so the positive real number √ x2 + y2 will be irrational: e.g. x = y = 1 =⇒ z = √ 2. However, a few integer solutions to x + y = z are familiar from high school algebra (and the SATs): e.g. (3, 4, 5), (5, 12, 13).
منابع مشابه
Parametrization of Pythagorean triples by a single triple of polynomials
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...
متن کاملHeight and Excess of Pythagorean Triples
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 ...
متن کاملDatasets on the statistical and algebraic properties of primitive Pythagorean triples
The data in this article was obtained from the algebraic and statistical analysis of the first 331 primitive Pythagorean triples. The ordered sample is a subset of the larger Pythagorean triples. A primitive Pythagorean triple consists of three integers a, b and c such that; [Formula: see text]. A primitive Pythagorean triple is one which the greatest common divisor (gcd), that is; [Formula: se...
متن کاملIndexing Properties of Primitive Pythagorean Triples for Cryptography Applications
This paper presents new properties of Primitive Pythagorean Triples (PPT) that have relevance in applications where events of different probability need to be generated and in cryptography.
متن کاملThe Modular Tree of Pythagorus
The Pythagorean triples of integers satisfying x + y = z have been studied and enumerated since Babylonian times. Since Diophantus, it has been known that this set of triples is related to the standard rational parameterization of the unit circle, ( t 2−1 t2+1 , 2t t2+1 ). The Pythagorean triple solutions, which are relatively prime, have the elementary and beautiful characterization as integer...
متن کاملAre monochromatic Pythagorean triples avoidable?
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 ...
متن کامل