Primitive pythagorean triples definition
The Pythagorean Theorem, arguably one of the bestknown results in mathematics, states that a triangle is a right triangle if and only if the sum of the squares of the lengths of two of its sides equals the square of the length of its third side. Closely associated with the Pythagorean Theorem is the concept of Pythagorean triples.A Pythagorean triple is an ordered triple (x, y, z) of three positive integers such that x 2 y 2 z 2. If x, y, and z are relatively prime, then the triple is called primitive. Let us first note the parity of x, y, and z in primitive triples, that is their values modulo 2. primitive pythagorean triples definition
A Pythagorean triple is a group of three integers (x, y, z) such that x2y2z2. When (x, y) are coprimes they are called primitive Pythagorean triples.
Primitive pythagorean triples definition free
Pythagorean Triples. A Pythagorean Triple is a set of positive integers, a, b and c that fits the rule: a 2 b 2 c 2. Example: The smallest Pythagorean Triple is 3, 4 and 5. The simplest way to create further Pythagorean Triples is to scale up a set of triples. Example: scale 3, 4, 5 by 2 gives 6, 8, 10. Which also fits the formula a 2 b
Primitive Pythagorean Triple Definition and Prop. Any Pythagorean triple with x, y, z mutually relatively prime; Any Pythagorean triple with two numbers sharing a factor can be reduced to a primitive triple. How to find all primitive Pythagorean triples.
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Such a triple is commonly written, and a wellknown example is. If is a Pythagorean triple, then so is for any positive integer k. A primitive Pythagorean triple is one in which a, b and c are coprime. A right triangle whose sides form a Pythagorean triple is called a Pythagorean triangle.
primitive triple and the original triple is a scalar multiple of this, so nding all Pythagorean triples is basically the same as nding all primitive Pythagorean triples. Our goal is to describe the primitive Pythagorean triples. We will be using di erent characterizations of primitive triples, as described in the following lemma. Lemma 1. 1.
A primitive Pythagorean triple is a Pythagorean triple (a, b, c) such that GCD(a, b, c)1, where GCD is the greatest common divisor. A right triangle whose side lengths give a primitive Pythagorean triple is then known as a primitive right triangle.
Definition. A primitive Pythagorean triple is a triple of positive integers that is a Pythagorean triple (i. e. , ) and either of these two equivalent conditions is satisfied: The are pairwise coprime integers. The are together coprime, i. e. , they have no factor common to both of them.
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