# Pythagorean Triples: Definition, Formula, List, Examples

## What are Pythagorean Triples

A **Pythagorean triple** is a set of three positive integers (a,b,c) that satisfy the equation a² + b² = c². The two smaller numbers, a and b, are the legs of a right-angled triangle, and the largest, c, is the hypotenuse — the side opposite the right angle.

The condition comes straight from the Pythagorean theorem: in any right triangle, the square of the hypotenuse equals the sum of the squares of the other two sides. A Pythagorean _triple_ is the special case where all three side lengths happen to be **whole numbers**.

The smallest and most-used triple is (3,4,5): 3² + 4² = 9 + 16 = 25 = 5².

### What makes three numbers a Pythagorean triple?

The test is a single equation. Square the two smaller numbers, add them, and check whether the result equals the square of the largest. If a² + b² = c² holds, the three numbers are a triple; if not, they are not.

## Primitive And Non-Primitive Triples

Pythagorean triples come in two kinds, and telling them apart matters.

- A **primitive Pythagorean triple** is one where the three numbers share no common factor larger than 1. Example: (3,4,5).
- A **non-primitive triple** is a multiple of a primitive one, so the three numbers share a common factor. Example: (6,8,10) is just (3,4,5) doubled.

## The Pythagorean Triples Formula

The most reliable way to generate triples is **Euclid's formula**. Pick any two positive integers m and n with m > n > 0. Then:

- a = m² - n²
- b = 2mn
- c = m² + n²

Here m and n are any two whole numbers you choose, with m larger than n.

A quick worked check. Take m=2, n=1:

- a = 2² - 1² = 4 - 1 = 3
- b = 2 × 2 × 1 = 4
- c = 2² + 1² = 4 + 1 = 5

That is (3,4,5).

## The List of Common Pythagorean Triples

| Primitive triple     | Check: a² + b² = c²  |
| --------------------- | -----------------  |
| (3,4,5)               | 9 + 16 = 25       |
| (5,12,13)            | 25 + 144 = 169    |
| (8,15,17)            | 64 + 225 = 289    |
| (7,24,25)            | 49 + 576 = 625    |
| (20,21,29)           | 400 + 441 = 841   |
| (9,40,41)            | 81 + 1600 = 1681  |

## Examples of Pythagorean Triples

### Example 1
**Verify that (8, 15, 17) is a Pythagorean triple.**

8² + 15² = 64 + 225 = 289, and 17² = 289. The two sides match.
**Final answer:** Yes, (8,15,17) is a Pythagorean triple.

### Example 2
**Is (6, 8, 11) a Pythagorean triple?**  
6² + 8² = 36 + 64 = 100, and 11² = 121. 
Since 100 ≠ 121, the equation fails.  
**Final answer:** No, (6,8,11) is not a Pythagorean triple.

### Example 3
**Generate a Pythagorean triple using Euclid's formula with m = 3 and n = 2.**

- a = 3² - 2² = 9 - 4 = 5
- b = 2 × 3 × 2 = 12
- c = 3² + 2² = 9 + 4 = 13
**Final answer:** The triple is (5,12,13).

### Example 4
**Create a non-primitive triple from (3, 4, 5).**

Using a factor of 4: (3×4, 4×4, 5×4) = (12,16,20)
**Final answer:** (12,16,20) is a non-primitive triple.

### Example 5
**A right triangle has legs of 9 and 40. Is its hypotenuse a whole number, and does this form a triple?**

9² + 40² = 81 + 1600 = 1681, and c = √1681 = 41. 
**Final answer:** Yes, (9,40,41) is a Pythagorean triple.

### Example 6
**The two legs of a right triangle are 20 and 21. Find the hypotenuse and state whether the triple is primitive.**

20² + 21² = 400 + 441 = 841, and c = √841 = 29.  
**Final answer:** The hypotenuse is 29, and (20,21,29) is a primitive triple.

## Why Pythagorean Triples Matter

Pythagorean triples exist because builders, navigators, and engineers needed right angles. A whole-number triple lets you measure a perfect 90° corner using nothing but a tape measure.

- **Construction:** Carpenters use the "3-4-5 rule" to square up foundations and walls.
- **Navigation and surveying:** Right-angle layouts rely on whole-number relationships.
- **Number theory:** Triples are important in mathematics, tying to Fermat's Last Theorem.

## The Mistakes Students Make Most Often

### Mistake 1: Adding the numbers instead of squaring them  
**Don't do this:** Check whether a+b=c. 
**Correct way:** The test is a² + b² = c².

### Mistake 2: Putting the largest number in the wrong place  
**Don't do this:** Test the squares incorrectly.
**Correct way:** The largest number is always the hypotenuse in the equation.

### Mistake 3: Confusing primitive with non-primitive  
**Don't do this:** Offer a non-primitive when a primitive triple is required.
**Correct way:** Check for a shared factor to ensure it's primitive.

## Conclusion

- A **Pythagorean triple** is three positive integers (a,b,c) with a² + b² = c².
- The largest number is always the hypotenuse; the two smaller ones are the legs.
- **Primitive triples** share no common factor; non-primitive triples are scaled copies.
- **Euclid's formula** generates triples from any two integers m > n.

- Common triples include (3,4,5), (5,12,13), (8,15,17), and (7,24,25).