Let a gaussian circle $C_R$ be any circle defined by the equation:

$$x^2+y^2 = R, (x,y) \in \mathbb{R}^2$$, where $R$ is the norm of a gaussian integer ($R=a^2+b^2, (a,b) \in \mathbb{Z}^2$). IF $R$ contains $n$ prime factors, it is not hard to show that $C_R$ contains $2^{n+2}$ integer points organized in $2^n$ classes of 4 elements. For example, with $R=65=5\times 13$ we have the only 4 integer points classes: $(\pm4,\pm7), (\pm8, \pm1), (\pm7,\pm4)$ and $(\pm1, \pm8)$.

We define the "non-abelian" addition law $\times$ on $C_R$ as follows:

$$(x_0, y_0) \times (x_1, y_1) = \frac{1}{R}\left(x_0\left(x_1^2-y_1^2\right)-2y_0x_1y_1, 2x_0x_1y_1+y_0 \left(x_1^2-y_1^2\right)\right),$$

the set of all rational points on $C_R$ forms a semigroup $G_R$, and if $P$ is an integer point on $C_R$ the set generated by repeated addition is dense on $C_R$.

Suppose now that $R$ factorization is unknown. Is it possible to find any **non-trivial** rational point on $C_R$?

I suspect the answer is no, otherwise we could easily factor large Gaussian integers. On the other hand, since rational points are dense on $C_R$, we have an infinity to choose from ...