3 problems

Let $p_1,\ldots,p_\ell\in\mathbb{Z}[t]$ be jointly intersective, meaning that for every $r\in\mathbb{N}$ there exists $n\in\mathbb{N}$ such that $r$ divides each of…

Polynomial-pattern conjecture. For any integer-valued polynomial $P(x)\in\mathbb{Q}[x]$ such that both $P(x)$ and $P(x)+x$ are nonconstant, there exists $t\in\mathbb{Z}$ such that

Polynomial Szemerédi-type conjecture. The set $S_F$ is syndetic. This is proposed as a polynomial strengthening of the preceding corollary, motivated by polynomial extensions of Sz…