4 problems

Let $f\in\mathbb{Z}[x]$ be a polynomial of degree $d\geq 3$ without roots of multiplicity $k$ or greater. An integer $a$ is free of $k$th powers if there is no integer $b>1$ such t…

Erdős's conjecture. The set $f(\mathbb{P})$ contains infinitely many $(d-1)$-free values.

Positive-density conjecture. This quantity has a positive density as $H\to\infty$.

Let $k\geq 2$ be an integer and let $f(x)\in\mathbb{Z}[x]$ be an irreducible polynomial. An integer is $k$-free if no prime $p$ satisfies $p^k\mid n$. A prime $p$ is a fixed $k$-th…