Pythagorean Triples Formula and Lists - physicscatalyst's Blog

Introduction to Pythagorean Triples

A Pythagorean triple is made up of three positive numbers, a, b, and c, so that $a^2 + b^2 = c^2$.
These numbers indicate the side lengths of a right-angled triangle, with ‘c’ representing the hypotenuse length.
Pythagorean triples are an extension of Pythagoras’ Theorem, which asserts that the square of the length of the hypotenuse in a right-angled triangle equals the sum of the squares of the other two sides.

How to generate Pythagorean Triples

There are several methods to generate Pythagorean triples, but one of the most common ways is by using Euclid’s formula

a = m² – n²
b = 2mn
c = m² + n²

Where:

a, b, and c are the side lengths of the right-angled triangle
m and n are any two positive integers with m > n > 0
m and n are coprime and both should not be odd numbers

Proof of Euclid Formula

$a^2 + b^2 = (m^2 -n^2)^2 + 4m^2n^2 =m^4 + n^4 + 2m^2n^2 =(m^2 + n^2)^2= c^2$

Important points

if a, b , c is a Pythagorean triplet, then k.a, k.b and k.c are considered as the Pythagorean triple.
6, 2.5, 6.5 are the sides of the right angle triangle but this is not Pythagorean triplet as it should be positive integers only
A primitive Pythagorean triplet has a coprime a, b, and c. (that is, they have no common divisor larger than 1)

Examples of Generating Triplets

Question 1: Generate a Pythagorean triple using Euclid’s formula with m = 3 and n = 2.

Solution: Apply Euclid’s formula with the given values of m and n:

a = m² – n² = 3² – 2² = 9 – 4 = 5
b = 2mn = 2 × 3 × 2 = 12
c = m² + n² = 3² + 2² = 9 + 4 = 13

So, the Pythagorean triple generated is (5, 12, 13)

Question 1: Generate a Pythagorean triple using Euclid’s formula with m = 2 and n = 1.

Solution: Apply Euclid’s formula with the given values of m and n:

a = m² – n² = 2² – 1² = 3
b = 2mn = 2 × 2 × 1 = 4
c = m² + n² = 2² + 1² = 5

So, the Pythagorean triple generated is (3, 4, 5)

How to generate Pythagorean Triples if one number is given

Case 1 : The given number is even ( m)
then
The Pythagorean Triples will be given by
m
$\frac {m^2}{2} -1$
$\frac {m^2}{2} +1$

Case 1 : The given number is odd ( m)
then
The Pythagorean Triples will be given by
m
$\frac {m^2}{2} -.5$
$\frac {m^2}{2} +.5$

Examples

If m =3, then the other will be
$\frac {m^2}{2} – .5= 4.5 -.5=4$
$\frac {m^2}{2} + .5= 4.5 +.5=5$
So triplet is (3,4,5)

Pythagorean Triples lists

Here is Primitive Pythagorean triples till 100

Frequently Asked Questions

Are all Pythagorean triples primitive?

No, not all Pythagorean triples are primitive. A primitive Pythagorean triple has side lengths that are coprime, meaning they share no common divisors other than 1. Non-primitive Pythagorean triples can be generated by multiplying all sides of a primitive triple by a common integer.

What are some common examples of Pythagorean triples?

Some common examples of Pythagorean triples are (3, 4, 5), (5, 12, 13), (7, 24, 25), and (8, 15, 17). These are primitive Pythagorean triples, as their side lengths share no common divisors other than 1.

Can a Pythagorean triple have an odd hypotenuse?

No, a Pythagorean triple cannot have an odd hypotenuse. The hypotenuse (c) will always be even since, according to Euclid’s formula, c = m² + n². Since the sum of two odd numbers is even, and the sum of two even numbers is also even, the hypotenuse will always be even.