8 problems

Bounded-gcd conjecture. For every Kurepa factorial sequence, $\mathcal{F}_n(a)$ is bounded above by $2$, with

The alternating-sum gcd conjecture. For all positive $m$ and $n$,

Let $(a(n))_{n\geq 1}$ be the EKG sequence defined by $a(1)=1$, $a(2)=2$, and, for $n\geq 3$, choosing the smallest unused natural number sharing a nontrivial common divisor with…

Let $(a(n))_{n\geq 1}$ be the EKG sequence defined by $a(1)=1$, $a(2)=2$, and, for $n\geq 3$, letting $a(n)$ be the smallest natural number not already in the sequence such that…

For each integer $n\geq 2$, let the left factorial $\mathbf{!n}$ be the sum of the factorials below $n$, and let $n!$ denote the usual factorial. Kurepa's conjecture. The two facto…

Let $(u_n)_{n\geqslant 0}$ be a Lucas sequence, and let $g_u(n)$ denote the gcd function associated with it. Assume that $\Delta_u\ne 1$. Then the maximal gcd conjecture asserts th…

Ailon–Rudnick conjecture. The set $\mathcal{A}_{F,G}$ is infinite.

Second binomial gcd infinitude conjecture. There are infinitely many positive integers $n$ such that