Has nontrivial solution in positive integers of a [diophantine equation](https://en.wikipedia.org/wiki/Diophantine_equation) as follows ?


$$x\_1^2+x\_2^2+x\_3^2+x\_4^2=y\_1^2+y\_2^2+y\_3^2+y\_4^2$$


Where trivial solutions are $x\_i=y\_j$.


Can you send me any reference for this question?



Has nontrivial solution in positive integers of a diophantine equation: $x_1^2+x_2^2+x_3^2+x_4^2=y_1^2+y_2^2+y_3^2+y_4^2$
There are many non-trivial solutions. By [Lagrange's four-square theorem](https://en.wikipedia.org/wiki/Lagrange%27s_four-square_theorem), *every* natural number can be represented as a sum of $4$ squares (with $0^2$ allowed), and [Jacobi's four-square theorem](https://en.wikipedia.org/wiki/Jacobi%27s_four-square_theorem) gives the number $r\_4(n)$ of representations of $n$. Note that $r\_4(n)$ double counts solutions, but since $r\_4(n)$ is arbitrarily large infinitely often, this clearly implies that there exist infinitely many $n$ with two distinct representations (without $0^2$). 


