Pythagorean Triples

A Pythagorean triple is a set of three positive integers (a, b, c) where the sum of the squares of the first two numbers equals the square of the third number: $$ a^2 + b^2 = c^2 $$

In other words, these three positive integers satisfy the Pythagorean theorem.

According to the Pythagorean theorem, in a right triangle, the square of the length of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the lengths of the other two sides.

So, if c is the length of the hypotenuse and a and b are the lengths of the legs of the right triangle, then the sum of the squares of the legs is equal to the square of the hypotenuse:

$$ a^2 + b^2 = c^2. $$

For example, the set of numbers (3, 4, 5) forms a Pythagorean triple because:

$$ 3^2 + 4^2 = 5^2 $$

$$ 9 + 16 = 25 $$

$$ 25 = 25 $$

Another example of a Pythagorean triple is (5, 12, 13)

$$ 5^2 + 12^2 = 13^2 $$

$$ 25 + 144 = 169 $$

$$ 169 = 169 $$

Yet another Pythagorean triple is (8, 15, 17)

$$ 8^2 + 15^2 = 17^2 $$

$$ 64 + 225 = 289 $$

$$ 289 = 289 $$

The set (7, 24, 25) is also a Pythagorean triple

$$ 7^2 + 24^2 = 25^2 $$

$$ 49 + 576 = 625 $$

$$ 625 = 625 $$

Pythagorean triples have been known since ancient times and are studied in number theory and geometry.