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SeventhSigma
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Given http://en.wikipedia.org/wiki/Pythagorean_quadruple it shows how to get primitive solutions, but what if I want even solutions that don't have coprime a, b, c, d?
Pythagorean quadruplets are sets of four positive integers (a, b, c, d) that satisfy the Pythagorean theorem, which states that in a right triangle, the square of the length of the hypotenuse is equal to the sum of the squares of the other two sides (a² + b² = c²).
Yes, there is a formula for generating Pythagorean quadruplets, known as the Euclid's formula. It states that if m and n are any two positive integers with m > n, then the quadruplet (a, b, c, d) = (m² - n², 2mn, m² + n², m² + 2mn + n²) is a Pythagorean quadruplet.
There are infinitely many Pythagorean quadruplets, as the Euclid's formula can generate an infinite number of them by choosing different values for m and n.
Yes, all Pythagorean quadruplets can be generated using the Euclid's formula. However, not all quadruplets generated by the formula will be unique, as some may have common factors.
Yes, there are other methods for generating Pythagorean quadruplets, such as using Pythagorean triples and adding an extra term, or using trigonometric identities. However, these methods may not be as efficient as the Euclid's formula.