18 problems

Let $a,b\in\mathbb{Z}$ be multiplicatively independent. Ailon–Rudnick's conjecture. One has … for infinitely many $n\geq 1$. This strengthens the preceding bound on the gcd of the…

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…

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

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

Let $k$ be an algebraically closed field of characteristic $0$, let $E/k(T)$ be an elliptic curve, and let $P,Q\in E(k(T))$ be independent points. Writing $x_R=A_R/D_R^2$ with copr…

Ailon–Rudnick conjecture. If $a$ and $b$ are multiplicatively independent, then

Let $A$ be an abelian variety of dimension at least $2$ defined over a number field $K$, let $P\in A(K)$, and assume that $\mathbb{Z}P$ is Zariski dense in $A$. Let $S$ be a finite…

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…

Fast-growth conjecture. Generalized divisor divisibility sequences should exhibit the fast-growth property whenever their associated Mahler measures are greater than $1$.

Zsigmondy finiteness conjecture. If

Growth conjecture.

Growth conjecture.

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…

Erdős's conjecture. The displayed limit is infinite. Stewart recently resolved this conjecture, so the claim is no longer open.

Let $L_n=(\alpha^n-\beta^n)/(\alpha-\beta)\in\mathbb{Z}$ be a Lucas sequence of the first kind, where $\alpha$ and $\beta$ are roots of a quadratic polynomial in $\mathbb{Z}[x]$. T…