Pythagorean Triples - Maple Help

Pythagorean Triples

Main Concept

A Pythagorean triple consists of three positive integers, $a$, $b$,  and $c$ such that ${a}^{2}+{b}^{2}={c}^{2}$.

These triples are usually denoted as $\left(a,b,c\right)$. The simplest and most common triple is $\left(3,4,5\right)$.

Euclid's formula can be used to generate a Pythagorean triple given an arbitrary pair of positive integers $m$ and $n$ where  :

$a={m}^{2}-{n}^{2}$

Primitive Triples(PPT)

If $a$, $b$, and $c$ are mutually prime or co-prime, the triple is known as a primitive. A primitive triple has many special properties such as:

 • .
 • $\frac{\left(c-a\right)\left(c-b\right)}{2}$ is always a perfect square.
 • At most one of $a$, $b$, $c$ is a square.
 • Exactly one of $a$, $b$ is odd; $c$ is odd.
 • Exactly one of $a$, $b$ is divisible by 3.
 • Exactly one of $a$, $b$ is divisible by 4.
 • Exactly one of $a$, $b$, $c$ is divisible by 5.
 • The area   is an even number.
 • By definition, $A$ is also congruent, that is, a positive integer which is the area of a right angled triangle with rational numbered side lengths.

Adjust the sliders or type positive integers in the boxes to change m and n and create the various Pythagorean triples.

Note: If  the computer will make . If $m\mathit{=}n$, no triangle can be formed.



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