# A family of Diophantine equations with no integer solutions but solutions modulo every integer

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).

• In your formulas the indices go $x_1,x_2,x^n$, some downstairs and some upstairs. I suppose that's a typo. Aug 1, 2021 at 16:53
• @Wojowu Fixing now. Thanks. Aug 1, 2021 at 17:06