# Pythagorean triples

A Pythagorean triple is a set of three positive integers that satisfies the equation: a^{2} + b^{2} = c^{2}.

In other words, if a, b, and c are positive integers where c is greater than a and b, and a^{2} + b^{2} = c^{2}, then a, b, and c are Pythagorean triples.

For example, 3, 4, and 5 form a Pythagorean triple since:

3^{2} + 4^{2} = 9 + 16 = 25 = 5^{2}

## Primitive Pythagorean triples

A primitive Pythagorean triple is a Pythagorean triple in which the three integers have no common divisor larger than 1.

The Pythagorean triple, 3, 4, 5, is the smallest triple integers that satisfies the Pythagorean Theorem; it is also a primitive Pythagorean triple because 3, 4, and 5 have no common divisors larger than 1. Some other primitive Pythagorean triples are:

5, 12, 13 |

7, 24, 25 |

8, 15, 17 |

9, 40, 41 |

11, 60, 61 |

## Non-primitive or reducible Pythagorean triples

Non-primitive Pythagorean triples are multiples of primitive Pythagorean triples. Multiplying the primitive triple 3, 4, 5 by 2 yields the non-primitive Pythagorean triple, 6, 8, 10, which has a common divisor of 2. We can confirm that this triple also satisfies the Pythagorean Theorem:

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

36 + 64 = 100

100 = 100

The table below shows three primitive Pythagorean triples and some of their multiples.

3, 4, 5 | 5, 12, 13 | 7, 24, 25 | |
---|---|---|---|

×2 | 6, 8, 10 | 10, 24, 26 | 14, 48, 50 |

×3 | 9, 12, 15 | 15, 36, 39 | 21, 72, 75 |

×4 | 12, 16, 20 | 20, 48, 52 | 28, 96, 100 |

×5 | 15, 20, 25 | 25, 60, 65 | 35, 120, 125 |

## Forming Pythagorean triples

There are many formulas that can be used to form a set of Pythagorean triples. One such formula involves the use of two positive integers, m and n, where m > n, such that:

a = m^{2} - n^{2}, b = 2mn, and c = m^{2} + n^{2}.

For example, if m = 4 and n = 3 then,

a = 4^{2} - 3^{2} = 16 - 9 = 7 |

b = 2 × 4 × 3 = 24 |

c = 4^{2} + 3^{2} = 16 + 9 = 25 |

Using the equation a^{2} + b^{2} = c^{2},

7^{2} +24^{2} = 49 + 576 = 625 = 25^{2}

satisfying the equation a^{2} + b^{2} = c, and confirming that 7, 24, 25 is a Pythagorean triple.