számhármasokkal

Számhármasokkal, often translated as "number triples" or "Pythagorean triples" in English, refers to sets of three positive integers that satisfy a specific mathematical relationship. This relationship is most famously embodied by the Pythagorean theorem, which states that in a right-angled triangle, the square of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the other two sides. Therefore, a számhármasokkal is a set of three positive integers (a, b, c) such that a² + b² = c².

The most well-known example of a számhármasokkal is (3, 4, 5), because 3² + 4² = 9 + 16 = 25, and 5² = 25. Other common examples include (5, 12, 13) and (8, 15, 17). These triples are fundamental in number theory and have applications in geometry, particularly in constructing right-angled triangles with integer side lengths.

Primitive számhármasokkal are those where the three integers have no common divisor greater than 1. The triple (3, 4, 5) is primitive. Any multiple of a primitive számhármasokkal also forms a számhármasokkal. For instance, multiplying (3, 4, 5) by 2 gives (6, 8, 10), and 6² + 8² = 36 + 64 = 100, while 10² = 100.

Euclid's formula provides a method for generating all primitive Pythagorean triples. For any two positive integers m and n, with m > n, m and n coprime and not both odd, the integers a = m² - n², b = 2mn, and c = m² + n² form a primitive Pythagorean triple.