4 problems

Consider a prime walk (PW) on the plane, and let $(x,y)$ denote a visited point. The number of visits to a point is its visit frequency. Prime-walk recurrence conjecture. In the li…

Let $A(N)$ denote the area covered by a prime walk up to the explored interval of length $N$, namely the number of unique lattice points $(x,y)$ visited by the walk. Area divergenc…

Monotonicity conjecture. The farthest distance reachable with a given step bound $k$ increases as $d$ decreases, subject to the constraint that $\mathcal{O}_{\mathbb{Q}(\sqrt{d})}$…

Moat-distance ordering conjecture. For a given step bound $k$, one can travel farther from the origin in $\mathbb{Z}[i]$ and $\mathbb{Z}[\omega]$ than in $\mathbb{Z}[\sqrt{-2}]$.