15 problems

Local-global conjecture for Apollonian bends. All sufficiently large integers satisfying certain congruence restrictions appear as the bends of some circle in the Apollonian circle…

Apollonian packing local-global conjecture. In any integral Euclidean Apollonian packing, all sufficiently large integers not ruled out by congruence conditions modulo $24$ are rep…

Let a primitive integral Apollonian packing be an integral Apollonian packing whose curvature quadruples have greatest common divisor $1$. An integer occur in the packing when it i…

Let an integral Apollonian packing be an Apollonian circle packing whose circle curvatures are all integers. An integer occur in such a packing when it is the curvature of one of i…

Haag–Kertzer–Rickards–Stange conjecture. The sporadic set $S({\mathcal P})$ is finite.

Graham–Lagarias–Mallows–Wilks–Yan density conjectures. The congruence obstruction is essentially the only obstruction, so the set of curvatures in a primitive integral Apollonian c…

Let ${\mathcal{P}^{\operatorname{full}}}$ be a primitive integral Apollonian packing, and let $C_{{\mathcal{P}^{\operatorname{full}}}}(X)$ be the number of circles of curvature at…

Let $C_{{\mathcal{P}^{\operatorname{pr}}}}(X)$ be the number of circles of curvature at most $X$ in a prime component ${\mathcal{P}^{\operatorname{pr}}}$, and let…

Let ${\mathcal{P}^{\operatorname{full}}}$ be an ambient primitive integral Apollonian packing. A prime component ${\mathcal{P}^{\operatorname{pr}}}$ is a component consisting of th…

Let ${\mathcal{P}^{\operatorname{full}}}$ be a primitive integral packing, and let $\Sigma$ be the union of the residue classes modulo $24$ that have representatives in…

Consecutive-curvature sequence conjecture. There is a sequence of consecutive tangent spheres

Curvature realization conjecture. The set of curvatures of $\Omega(\mathcal{B}_{\mathcal{R}^4})$ is $\mathbb N$.

Local-global principle. Every sufficiently large admissible integer occurs as the curvature of a sphere in the packing.

Let $dim=3$, $4$, or $5$. Let $R(\Q)$ be either an imaginary quadratic field (if $dim=3$), a rational, definite quaternion algebra equipped with an orthogonal involution $\ddagger$…

Fuchs–Sanden conjecture. All primitive integral Apollonian packings can be described in terms of congruence conditions modulo $24$.