# Pythagorean Triples

A "Pythagorean Triple" is a set of positive integers, **a**, **b** and **c** that fits the rule:

a^{2} + b^{2} = c^{2}

### Example: The smallest Pythagorean Triple is 3, 4 and 5.

Let's check it:

3^{2} + 4^{2} = 5^{2}

Calculating this becomes:

9 + 16 = 25

Yes, it is a Pythagorean Triple!

## Triangles

When a triangle's sides are a Pythagorean Triple it is a right angled triangle.

See Pythagoras' Theorem for more details.

### Example: The Pythagorean Triple of 3, 4 and 5 makes a Right Angled Triangle:

Here are two more Pythagorean Triples:

5, 12, 13 | 9, 40, 41 | |

5^{2} + 12^{2} = 13^{2} | 9^{2} + 40^{2} = 41^{2} | |

25 + 144 = 169 | (try it yourself) |

And each triangle has a right angle!

## List of the First Few

Here is a list of the first few Pythagorean Triples (**not** including "scaled up" versions mentioned below):

(3, 4, 5) | (5, 12, 13) | (7, 24, 25) | (8, 15, 17) |

(9, 40, 41) | (11, 60, 61) | (12, 35, 37) | (13, 84, 85) |

(15,112,113) | (16, 63, 65) | (17,144,145) | (19,180,181) |

(20, 21, 29) | (20, 99,101) | (21,220,221) | (23,264,265) |

(24,143,145) | (25,312,313) | (27,364,365) | (28, 45, 53) |

(28,195,197) | (29,420,421) | (31,480,481) | (32,255,257) |

(33, 56, 65) | (33,544,545) | (35,612,613) | (36, 77, 85) |

(36,323,325) | (37,684,685) | ... infinitely many more ... |

## Scale Them Up

The simplest way to create further Pythagorean Triples is to scale up a set of triples.

### Example: scale **3, 4, 5** by 2 gives **6, 8, 10**

Which also fits the formula a^{2} + b^{2} = c^{2}:

6^{2} + 8^{2} = 10^{2}

36 + 64 = 100

If you want to know more about them read Pythagorean Triples - Advanced