Pythagorean Triples
Discover integer solutions of the Pythagorean theorem and how to generate them
Overview
A Pythagorean triple is a set of three natural numbers \((a, b, c)\) that can be side lengths of a right triangle.
The triple must satisfy the Pythagorean equation:
If \(a\), \(b\), and \(c\) have no common divisor other than 1, the triple is called primitive.
\((3, 4, 5)\) is the smallest primitive Pythagorean triple.
Angle bisector
Median
Altitude and the Orthocenter
Triangle bisector
Pythagorean theorem
Pythagorean triples
Circle
Square
Rectangle
Rhombus
Parallelogram
Trapezoid
Primitive and Non-Primitive Triples
Primitive
No common divisor greater than 1
Examples: \((3,4,5)\), \((5,12,13)\), \((8,15,17)\)
Non-Primitive
All numbers share a common divisor greater than 1
Examples: \((15,20,25)\), \((15,36,39)\)
Every non-primitive triple is a multiple of a primitive one.
Generation of Pythagorean Triples
A standard way to generate triples uses two integers \(m\) and \(n\), with \(m > n\):
Euclid's formulas
\(\displaystyle b = 2mn\)
\(\displaystyle c = m^2 + n^2\)
- Choose integers \(m > n > 0\)
- If \(m\) and \(n\) are coprime and not both odd, the triple is primitive
- Otherwise, the result is a non-primitive triple
Example with \(m=2\), \(n=1\)
This gives the triple \((3,4,5)\).
More mathematical details are available at Wikipedia.
Key Points
- Pythagorean triples are integer solutions of \(c^2 = a^2 + b^2\)
- Primitive triples have greatest common divisor \(\gcd(a,b,c)=1\)
- \((3,4,5)\) is the smallest primitive triple
- Euclid's formulas generate infinitely many triples
- Every non-primitive triple is a scaled primitive triple
|
|
|
|