Historical context of the quadruple $(x,y,z,t)$ where $x^2+y^2=z^2+t^2$

Question

Recently my study is related to an equation $x^2+y^2=z^2+t^2$. I have learned that Euler studied about numbers which can be represented as a sum of two squares in two different way. But I do not know if there is anyone who studied this equation before him.

So I would like to ask if there is a name and the historical context of the quadruple $(x,y,z,t)$ so that $x^2+y^2=z^2+t^2$.

Answer 1

Any Pythagorean triple that has a hypotenuse that factors into $2$ distinct primes, to any power will have two sets of $(A,B)$ pairs where $A^2+B^2=C^2$. For example $65=5\times13$ so $$33^2+56^2=65^2\quad 63^2+16^2=65^2\\ \implies 33^2+56^2=63^2+16^2 $$

There are infinite numbers of such triple pairs. There will be $2^{n-1}$ primitive triples for any given hypotenuse where $(n)$ is the number of its unique prime factors. Some will have more triples than this but the extras will be non-primitive.

The only historical context I can think of is the development of sums of squares representing integers whether they are squares or not. Some studies on the subject are here and here.

Answer 2

It is well known since longtime ago that the complete solutions of the equation $x^2+y^2=z^2+t^2$ is given by the following identity in four parameters $r,s,X,Y$

$$(rX+sY)^2+(rY-sX)^2=(rX-sY)^2+(rY+sX)^2$$