pitagorasztriplák

Pitagorasztriplák are sets of three positive integers, denoted as (a, b, c), that satisfy 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. Mathematically, this relationship is expressed as a² + b² = c².

These integer solutions are fundamental in number theory and have applications in geometry and cryptography. The most basic and well-known example is the (3, 4, 5) triplet, as 3² + 4² = 9 + 16 = 25, and 5² = 25. Other common primitive Pythagorean triplets include (5, 12, 13), (8, 15, 17), and (7, 24, 25). A primitive Pythagorean triplet is one where the three integers are coprime, meaning their greatest common divisor is 1.

There are systematic methods to generate all Pythagorean triplets. One such method involves using Euclid's formula, which states that for any two positive integers m and n with m > n, the integers a = m² - n², b = 2mn, and c = m² + n² form a Pythagorean triplet. If m and n are coprime and one of them is even, then the generated triplet will be primitive. By multiplying a primitive triplet by any positive integer, other non-primitive triplets can be generated. For instance, multiplying (3, 4, 5) by 2 yields (6, 8, 10), which is also a Pythagorean triplet since 6² + 8² = 36 + 64 = 100, and 10² = 100.