The Core Concept
A Pythagorean triple is a set of three positive integers which satisfy Pythagoras' theorem. This means if you square the two smaller numbers and add them, you get the square of the largest number ($a^2 + b^2 = c^2$).
Primitive vs. Scaled Triples
A primitive triple is one where the three numbers have no common factors, meaning they are coprime.
You can generate new triples by multiplying a primitive triple by a common factor. However, these are just scaled-up versions. For example, 60, 80, 100 is not primitive because it is $20\times$ the 3, 4, 5 triple.
Interactive Triple Checker
Enter any three integers to see if they form a Pythagorean Triple, and whether they are primitive or scaled!
Euclid's Generator Challenge
Ancient mathematician Euclid proved that we can generate Pythagorean triples using two "seed" integers, $m$ and $n$, with his famous formula:
$a = m^2 - n^2$, $\quad b = 2mn$, $\quad c = m^2 + n^2$
Step 1: Choose your Seeds
- $m$ must be strictly greater than $n$ ($m > n$).
- Both must be positive integers ($n > 0$).
- To guarantee a PRIMITIVE triple: They must share NO common factors (they must be coprime), and exactly one of them must be even while the other is odd.
Step 2: Calculate the Sides
You chose $m = $ and $n = $. Now, use Euclid's formula to find $a, b,$ and $c$.
Alternative Methods
Pythagoras' Own Method
Before Euclid, the Greek philosopher Pythagoras had his own clever method to generate a triple starting with any odd number $a \ge 3$.
- Pick an odd number $a$.
- Calculate $b = \frac{a^2 - 1}{2}$
- Calculate $c = \frac{a^2 + 1}{2}$
Step 1: Pick an Odd Number
Step 2: Calculate b and c
You chose $a = $. Now calculate $b$ and $c$ based on Pythagoras' rules.
The Naive Scale Method
Once you have one primitive triple, you instantly have an infinite number of triples. Just multiply all three sides by the exact same integer $k$.
Let primitive = (3, 4, 5)
Multiply by $k=2$: (6, 8, 10)
Multiply by $k=10$: (30, 40, 50)
Multiply by $k=50$: (150, 200, 250)
Mastery Quiz
1. What defines a Pythagorean Triple? (Type A, B, or C)
- A) Any three odd integers.
- B) Three positive integers where $a^2 + b^2 = c^2$.
- C) Any three numbers that can form a triangle.
2. A primitive triple has no common factors among all three numbers. (True/False)
3. Using Euclid's formula, if $m=4$ and $n=1$, what is the hypotenuse ($c$)?
4. Is the triple 15, 20, 25 a primitive triple? (Yes/No)
5. The triple (27, 36, 45) is a scaled version of (3, 4, 5). What is the scale factor $k$?
6. Using Pythagoras' Odd Method: If side $a=7$, what is the length of side $b$? Hint: $(a^2-1)/2$
7. In Euclid's formula, to guarantee a primitive triple, $m$ and $n$ must be: (A, B, or C)
- A) Both odd numbers.
- B) Coprime, and exactly one of them is even.
- C) Both even numbers.
8. Challenge: If $m=5$ and $n=3$, what is the value of $a$ generated? ($a=m^2-n^2$)