4 problems

Bugeaud's conjecture. The Hurwitz constant of any real quadratic irrational is at most

Minimal-period uniqueness conjecture. For each prime $p$, the periodic part of the continued fraction of such a minimizing $\xi$ is unique up to reversal. Moreover,

Maximum conjecture. The quantity $\min \mathcal{L}_p$ attains a maximum at $p=67$, and

Lowest-point algebraicity conjecture. For all positive integers $n$, the lowest point $\min \mathcal{L}_n$ is an isolated point of $\mathcal{L}_n$ and is the irrational square root…