Pythagorean quintuple $x^2+y^2+z^2+w^2=u^2$ has infinite solutions. Is it difficult to find the general formula of its integer solution?

Question

Pythagorean quintuple equation $x^2+y^2+z^2+w^2=u^2$ has the infinite set of integer solutions, but it seems that no general solution formula has been published. There are methods to give more solutions, but is there a general formula for the solution? If we consider the Pythagorean multi-tuple equation, $x_ 1^2 + x_ 2^2 + \cdots + x_n ^ 2={x ^ 2} $, what is the maximum value of n, then we can find the general formula of its integer solution?

Answer 1

You have a known generalization of the Pythagorean triples valid for $n\ge2$. This is for your five unknowns $x,y,z,w,u$ the following identity with four parameters: $$(r^2+s^2+t^2+w^2)^2=(w^2-r^2-s^2-t^2)^2+(2rw)^2+(2sw)^2+(2tw)^2$$