7 problems

Let $t$ be a positive integer, let $S=\{x_1,x_2,\ldots,x_n\}$ be a GCD-closed set of positive integers, and let the power LCM matrix be the matrix … The Hong conjecture. There exis…

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 $[S]$.

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…