Pythagorean Triples
Main Concept
A Pythagorean triple consists of three positive integers, , , and such that .
These triples are usually denoted as . The simplest and most common triple is .
Euclid's formula can be used to generate a Pythagorean triple given an arbitrary pair of positive integers and where :
Primitive Triples(PPT)
If , , and are mutually prime or co-prime, the triple is known as a primitive. A primitive triple has many special properties such as:
.
is always a perfect square.
At most one of , , is a square.
Exactly one of , is odd; is odd.
Exactly one of , is divisible by 3.
Exactly one of , is divisible by 4.
Exactly one of , , is divisible by 5.
The area is an even number.
By definition, 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 , no triangle can be formed.
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