Question 46
It is known that the equations $x^4 + y^4 + z^4 = t^4$ and $x^4 + y^4 + z^4 = 2t^4$ (with unknowns $x$, $y$, $z$, $t$) have solutions in $\mathbb{N}^*$.
a) Find the smallest natural number $k$ such that the equation
$$
x^4 + y^4 + z^4 = k t^4
$$
has no solutions in $\mathbb{N}^*$.
b) Prove that for every $n \in \mathbb{N}$, $n \ge 3$ there exists $N \in \mathbb{N}$ such that $N$ can be written as the sum of two, three, four, ..., $n$ fourth powers of positive integers.
a) Find the smallest natural number $k$ such that the equation
$$
x^4 + y^4 + z^4 = k t^4
$$
has no solutions in $\mathbb{N}^*$.
b) Prove that for every $n \in \mathbb{N}$, $n \ge 3$ there exists $N \in \mathbb{N}$ such that $N$ can be written as the sum of two, three, four, ..., $n$ fourth powers of positive integers.