9 problems

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…

Let $b$ be an integer such that $-b$ is not an integer square, and define … Natural-density conjecture. As $x$ tends to infinity, … This would identify the natural density of indic…

Let $\beta$ be a nonzero integer and define the sequence $P=(P_n)$ by … A term $P_n$ has a primitive divisor if it has a prime divisor that divides no earlier nonzero term of the s…

Let $(A_n)$ be a sequence of integers. An integer $d>1$ is a primitive divisor of $A_n$ if $d\mid A_n$ and $(d,A_m)=1$ for every nonzero term $A_m$ with $m<n$. For $b\in\mathbb{Z}$…

Let $K$ be a number field, let $\mathcal{O}_K$ be its ring of algebraic integers, let $\varphi(z)\in K(z)$ be a rational function of degree $d\geqslant 2$, and let $\alpha\in K$ be…

Let $E_a$ be the elliptic curve considered above, let $P$ be a non-torsion point in $E_a(\mathbb{Q})$, and let $B_n(E_a,P)$ denote the associated elliptic divisibility sequence. Pr…

Dynamical Zsigmondy-set conjecture. With appropriate conditions on $f$ and $\alpha$ to rule out trivial counterexamples, the dynamical Zsigmondy set associated to $(f,\alpha)$ is a…

For each positive integer $i$, consider the integer $2^i-3$. A primitive prime divisor of $2^d-3$ is a prime divisor that divides none of the earlier terms $2^i-3$ with $i<d$. Prim…

Let $K$ be a number field, let $\phi(t)\in K(t)$ have degree $d\geq 2$, and let $x_{n+1}=\phi(x_n)$ with $x_0=0$. A prime factor of the numerator of $x_n$ is primitive if it does n…