11 problems

Bourque–Ligh conjecture. The LCM matrix $[S]$ is invertible for every such gcd-closed set $S$.

Zhu–Chen–Hu conjecture. When $a=b$, the three factorizations $(S^a)\mid(S^b)$, $(S^a)\mid[S^b]$, and $[S^a]\mid[S^b]$ in the corresponding integer matrix ring hold if and only if…

Zhao–Chen–Hong conjecture. If $\max_{x\in S}|G_S(x)|\geq 4$, then $[S^e]$ is divisible by $(S^e)$ in $M_n(\mathbb Z)$ if and only if $S$ satisfies condition $\mathcal C$.

Altinsik–Yildiz–Keskin conjecture. If $\max_{x\in S}|G_S(x)|\geq 2$ and $S$ does not satisfy condition $\mathcal M$, then $(S)$ does not divide $[S]$.

Zhao's conjecture. If $m\geq 4$ and $n<\binom{m}{2}+m+2$, then $(S^e)$ does not divide $[S^e]$ in $M_n(\mathbb Z)$.

Let $\mathcal{N}\subset\mathbb{N}^+$ be a finite GCD-closed set, meaning that $\operatorname{gcd}(m,n)\in\mathcal{N}$ whenever $m,n\in\mathcal{N}$, and suppose that $\mathcal{N}$ c…

Let $S=\{x_1,\ldots,x_n\}$ be an odd-GCD-closed set, and let $f$ be a completely multiplicative function, meaning $f(ab)=f(a)f(b)$ for positive integers $a,b$. If $f$ is strictly m…

Let $S=\{x_1,\ldots,x_n\}$ be an odd-GCD-closed set, meaning that $S$ consists of odd positive integers and is closed under greatest common divisors. For a real exponent $\alpha$,…

Let $S$ be a GCD-closed set of odd positive integers, meaning that every element of $S$ is odd and the greatest common divisor of any two elements of $S$ belongs to $S$. Let…

Let $S=\{x_1,\ldots,x_n\}$ be a GCD-closed set of odd positive integers, and let $[[x_i,x_j]^r]$ be its power LCM matrix with nonzero exponent $r$. Hong's conjecture. Every such po…

Let $S=\{x_1,x_2,\ldots,x_n\}$ be a GCD-closed set of positive integers, let $r\neq0$, and let $[[x_i,x_j]^r]$ denote its power LCM matrix. Shen's conjecture. The matrix…