# Pythagorean triples

A Pythagorean triple is a set of three positive integers that satisfies the equation: a2 + b2 = c2.

In other words, if a, b, and c are positive integers where c is greater than a and b, and a2 + b2 = c2, then a, b, and c are Pythagorean triples.

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

32 + 42 = 9 + 16 = 25 = 52

## 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:

62 + 82 = 102

36 + 64 = 100

100 = 100

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

3, 4, 55, 12, 137, 24, 25
×26, 8, 1010, 24, 2614, 48, 50
×39, 12, 1515, 36, 3921, 72, 75
×412, 16, 2020, 48, 5228, 96, 100
×515, 20, 2525, 60, 6535, 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 = m2 - n2, b = 2mn, and c = m2 + n2.

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

 a = 42 - 32 = 16 - 9 = 7 b = 2 × 4 × 3 = 24 c = 42 + 32 = 16 + 9 = 25

Using the equation a2 + b2 = c2,

72 +242 = 49 + 576 = 625 = 252

satisfying the equation a2 + b2 = c, and confirming that 7, 24, 25 is a Pythagorean triple.