A proof is given that the system in the title has infinitely many solutions of the form $a_1+a_2$, where $a_1$ and $a_2$ are rational numbers.

The two related Diophantine equations: $X_1^4+4X_2^4=X_3^8+4X_4^8$ and $Y_1^4=Y_2^4+Y_3^4+4Y_4^4$, have infinitely many nontrivial, primitive integral solutions. We give two parametric solutions, one for each of these equations.

In p. 219 of R.K. Guy’s Unsolved Problems in Number Theory, 3rd edn., Springer, New York, 2004, we are asked to prove that the Diophantine equation $x^n+y^n=n!z^n$ has no integer solutions with $n\in {\mathbb{N}}_{+}$ and $n>2$. But, contrary to this expectation, we show that for $n=3$, this equation has infinitely many primitive integer solutions, i.e. the solutions satisfying the condition $gcd(x,y,z)=1$.