That the curvatures are integers is proved (noticed by Soddy in 1936 as "a fairly straightforward consequence of Descartes’ equation", as the article mentions),

That the curvatures are integers is proved (noticed by Soddy in 1936 as "a fairly straightforward consequence of Descartes’ equation", as the article mentions), and a lot more is also proved about the integers' distribution; the conjecture specifically is about whether every sufficiently large “admissible (passing local obstructions) integer is the curvature of some circle in the gasket” — see the second page of https://arxiv.org/abs/1205.4416 https://arxiv.org/abs/1205.4416, or indeed the second page of the paper the article is about (https://arxiv.org/abs/2307.02749 https://arxiv.org/abs/2307.02749), which puts it even more concretely: > Conjecture 1.1 ([GLM+03, FS11]). Let A be a primitive Apollonian circle packing containing curvatures equivalent to r (mod 24). The set of positive integers x ≡ r (mod 24) not occurring in A is finite. which they disprove with: > Theorem 1.3. There exist infinitely many primitive Apollonian circle packings for which the number of missing curvatures up to N is Ω(√N). In particular, the local-global conjecture is false for these packings.