3 problems

The $p$-adic Jacobi–Perron algorithm is started with an $m$-tuple of $\mathbb Q$-linearly dependent numbers. Finite-or-periodic conjecture. The algorithm is finite or periodic. Thi…

Refinement conjecture. Every possible configuration of linear dependency is a refinement of one of such cases.

Let $E$ be an elliptic curve over $\mathbb{Q}$ of rank $r\geq 1$, let $\Gamma$ be a subgroup of $E(\mathbb{Q})$ of rank $s<r$, and let $p_n$ denote the $n$th prime. For each…