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=m2−n2
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 c−ac−b2.
c−ac−b2 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 = n+1. If m=n, no triangle can be formed.
m =
n =
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