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](https://mathoverflow.net/questions/47442/diophantine-equation-with-no-integer-solutions-but-with-solutions-modulo-every) and [here](https://mathoverflow.net/questions/2779/proof-of-no-rational-point-on-selmers-curve-3x34y35z3-0). A nice proof that the curve always have solutions for all $p$-adics is in this writeup by [Kevin Buzzard](https://www.ma.imperial.ac.uk/%7Ebuzzard/maths/teaching/10Aut/M4P32/exsht3.pdf). 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).



A family of Diophantine equations with no integer solutions but solutions modulo every integer
The answer is no. Heath-Brown has shown every cubic form over the integers in at least 14 variables represents zero nontrivially. The Wikipedia article on [Hasse principle](https://en.wikipedia.org/wiki/Hasse_principle#Cubic_forms) contains references as well as related results for forms in higher odd degrees.


