Pythagorean Triples - Maple Help
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Pythagorean Triples

Main Concept

A Pythagorean triple consists of three positive integers, a, b,  and c such that a2+b2=c2.


These triples are usually denoted as a,b,c. The simplest and most common triple is 3,4,5.


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



b=2 mn

c  =m2+n2

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:


a+b = c + 2 cacb2.


cacb2 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  A = ab2 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 m< n the computer will make m &equals; n&plus;1. If m&equals;n, no triangle can be formed.

m &equals; 

n &equals; 

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