3 problems

Let $q\in\mathcal{Q}_n$, and let $G_0(\operatorname{\mathbb{Q}}^n,q)$ be the connected component containing $\operatorname{\textbf{0}}$ in the distance graph associated with $q$. A…

Nonexistence conjecture for perfect Heron triangles. No perfect triangle exists.

Uniqueness conjecture. Up to scaling, this is the unique such tetrahedron with only one rational face area. The source gives this as a conjecture; no resolution is supplied here.