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 :

 

a=m2n2

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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