9 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…

Let $a_1G_1,\ldots,a_kG_k$ be pairwise disjoint left cosets in a group $G$, where each subgroup $G_i$ has finite index. Sun's finite-coset index conjecture. There are indices $i<j$…

Fibonacci squared-sum gcd conjecture. For all even $k$,

Digit-statistics gcd conjecture. For every positive integer $n$,

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…

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

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…