13 problems

Let $f\in\mathbb{Z}[T]$ be primitive and squarefree, let $\mathcal{A}=a+N\mathbb{Z}\subseteq\mathbb{Z}$ be an arithmetic progression, and let…

Let $F$ be an integer polynomial in $n$ variables, and order its integer points by a height function. For each prime number $p$, consider the density of $p$-adic points…

Let $P\in\mathbb{Z}[X_{1},\ldots,X_{n}]$ have total degree $d\geq 2$. For $m>1$, define … Given $B_j\in\mathbb{R}$ with $B_j\geq1$ and $B=\prod_{j=1}^{n}[0,B_j]\cap\mathbb{Z}^{n}$,…

Let $f(t)\in\mathbb{Z}[t]$ be a separable polynomial of degree $\operatorname{deg} f\geq 1$. Let $\mu$ denote the Möbius function, let $\rho_f(m)$ denote the number of residue clas…

Let $f(t)\in\mathbb{Z}[t]$ be a separable polynomial such that $f(t)\ne a(t)b(t)^2$ and $\operatorname{deg} f\geq 1$. Let $\mu$ denote the Möbius function and let $\rho_f(m)$ denot…

Squarefree-values conjecture. Then $f(n)/B'$ is squarefree for infinitely many values of $n$.

Squarefreeness conjecture. For every $p$ (and the fixed $a$), the integer $m=a^2p^{2\rho}+1$ is squarefree for infinitely many integers $\rho\geq 2$. This is used to obtain an infi…

Let $f\in\mathbb{Z}[x_1,\ldots,x_h]$ be primitive and squarefree, and let $\mathbb{Z}^h(X)=\{{\rm v}\in\mathbb{Z}^h:|{\rm v}|\leq X\}$. Squarefree conjecture, alternative version.…

Let $h\in\{1,2\}$, let $f\in\mathbb{Z}[x_1,\ldots,x_h]$ be primitive and squarefree, and let $\mathbb{Z}^h(X)=\{{\rm v}\in\mathbb{Z}^h:|{\rm v}|\leq X\}$. Define…

Squarefree-value residue class conjecture. For all but finitely many primes $p$, the set $\mathcal S(f)$ contains infinitely many elements from every nonzero residue class modulo…

For each prime $p$, let ${\mathcal C}_p$ be the set defined in the paper that records the relevant sporadic roots or residue classes. Average-size conjecture for ${\mathcal C}_p$.…

Let $S$ be the set of positive integers $n$ for which $n^n+(-1)^n(n-1)^{n-1}$ is squarefree. Squarefree-value density conjecture. The set $S$ has density … correct to that many dec…

Squarefree-value conjecture. As $X\to\infty$,