Selmer's curve is the equation $3x^3 +4y^3 +5z^3=0$. This equation is famous for having non-trivial solutions in every completion of $\mathbb{Q}$ but only having the trivial solution in the rationals. This curve has been discussed on Mathoverflow before such as here and here. A nice proof that the curve always have solutions for all $p$-adics is in this writeup by Kevin Buzzard. I have two questions related to this curve.

**Question I:** How much worse can we get for cubics if the number of variable is increased. That is:

For every $n \geq 3$ is there a list of non-zero integers $a_1, a_2 \cdots a_n$ such that the equation $$a_1x_1^3 +a_2x_2^3 \cdots a_n x_n^3 =0$$ has solutions in every completion of $\mathbb{Q}$ but no non-trivial integer solutions?

**Question II:** can we make a family of such equations which is nested? That is is there a sequence of non-zero integers $a_1, a_2, a_3 \cdots $ such that for any $n \geq 3$ the equation $$a_1x_1^3 +a_2x_2^3 \cdots a_n x_n^3 =0$$ has solutions in every completion of $\mathbb{Q}$ but no non-trivial integer solutions? And if so, can we take $a_1=3$, $a_2=4$ and $a_3=5$ (that is using Selmer's curve as the start of our family).