9 problems

Geometric divisibility sequence conjecture. The geometric divisibility sequence $(D_{n\mathcal{P}})_{n\geq 1}$ corresponding to $\mathcal{P}$ satisfies

Let $\mathcal{G}/\mathbb{Z}$ be a group scheme, let $\sigma_P\in\mathcal{G}(\mathbb{Z})$, and write … for its generic fiber. Assume that $G$ is an irreducible commutative algebraic…

Let $\alpha \in \bar{\Z}$, let $K=\Q(\alpha)$, and let $d_k(\alpha)$ denote the largest positive integer such that … Assume that either (a) $[\Q(\alpha^r):\Q]\geq 3$ for every…

Growth conjecture.

Primitive divisor conjecture. The term $a_n$ has a primitive prime divisor for all but finitely many $n$. Moreover, there exists a positive integer $M$ such that, whenever $a_n$ do…

Square-freeness conjecture. If $n$ is square-free, then $b_n$ is square-free.

Gcd conjecture. If $(b_m,b_n)>1$, then

For an integer $a$ and a positive integer $r$, define … where $\operatorname{ord}_p(a)$ is the multiplicative order of $a$ modulo the prime $p$. Erdős's conjecture. For every…

Exact valuation conjecture. For the relevant integers $m$ and $l$, one has