5 problems

Let $\sG$ be the fixed primitive integral Apollonian gasket, let $\sB=\sB(\sG)$ be the set of its circle curvatures, … and call an integer represented if it lies in $\sB$. Let…

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

Let … and consider its orbit of the indicated vector. Explicit $\Psi_1$ numerator conjecture. Every non-square integer $n>10569$ appears as a numerator in the orbit…

Define … The table of congruence and reciprocity obstructions specifies the stated restrictions on numerators and denominators in these orbits. Local-global conjecture for $\Psi_1$…

Let $\mathcal{A}\subseteq\mathbb{N}$ be finite, let $v$ be the primitive vector from the associated orbit, and let $L$ be a linear functional. Define … and let…