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 $x^2+y^2=z^2+t^2$.
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.
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 $r,s,X,Y$
$$(rX+sY)^2+(rY-sX)^2=(rX-sY)^2+(rY+sX)^2$$
