- #1
Dragonfall
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Which squares are expressible as the sum of two squares? Is there a simple expression I can write down that will give me all of them? Some of them? Parametrization of the pythagorean triples doesn't seem to help.
Pythagorean Triples are three positive integers (a, b, c) that satisfy the Pythagorean Theorem, a² + b² = c². In simpler terms, it is a set of three numbers that can form a right triangle.
One way to find Pythagorean Triples is by using the formula a = m² - n², b = 2mn, c = m² + n², where m and n are positive integers with m > n. Another method is by using the Euclidean algorithm to generate primitive Pythagorean Triples, which can then be multiplied by a constant to get non-primitive triples.
Pythagorean Triples have been studied for centuries and have many applications in mathematics, physics, and engineering. They are also used in solving problems related to right triangles, such as finding missing side lengths or angles.
Yes, there are several patterns and rules that can help in finding Pythagorean Triples. For example, if one of the numbers in a triple is a multiple of 3, then the other two numbers will also be multiples of 3. Additionally, all Pythagorean Triples can be classified into three categories: primitive, non-primitive, and special.
Yes, Pythagorean Triples can be extended to more than three numbers, known as Pythagorean n-tuples. These are sets of n positive integers that satisfy the Pythagorean Theorem, such as (3, 4, 5, 6) or (5, 12, 13, 16, 20).