C.2 Python Output:

0.2510066561587017
0.1254917624198063 0.6768927381439971 0.5345668734285826
2.0001843253982816 0.3708219072447848 0.4695514605100869

The first figure is an approximation of the density of integer unit circles centred at (−1,0)(-1,0), (0,0)(0,0) and (1,0)(1,0).

Using the formula for the density of intersections of two integer unit circles centred apart from each other, we guessed at potential formulae for this. In order, the results give approximations of:

∏p∈ℙ(1−3p2);\prod_{p\in{\mathbb{P}}}\left(1-\frac{3}{p^{2}}\right);
∏p∈ℙ(1−2p3);\prod_{p\in{\mathbb{P}}}\left(1-\frac{2}{p^{3}}\right);
∏p∈ℙ(1−3p3).\prod_{p\in{\mathbb{P}}}\left(1-\frac{3}{p^{3}}\right).

The next output is the ratio between the approximated density the first output and the computed approximations of the above products (in the same order). Notice that none of these are particularly close to 1.