## On constructing free action of the cyclic group $\Bbb Z/3\Bbb Z$ on $S^n \times \cdots \times S^n$($n$ is odd)

Can we construct a free action of the cyclic group $$\Bbb Z/3\Bbb Z$$ on $$S^n \times \cdots \times S^n$$($$n$$ is odd) without multiplying any coordinate by $$e^{2\pi i/3}$$? In other words, do I Need to multiply at least one coordinate by a 3rd root of unity to get a free action of $$\Bbb Z/3\Bbb Z$$? I have tried to construct such examples but could not find any.

Thank you so much in advance.

## Hypothesis $\cdots$

for $$n, p\in\mathbb{N}(p-prime\>number)$$ there exist $$m\in\mathbb{N}$$ for $$\p^m\equiv m\pmod n$$