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, 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 = 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.