# A coprimality condition on consecutive values of polynomials

## TL;DR

This paper proves that for quadratic or cubic polynomials, beyond a certain threshold, there are infinitely many sequences of consecutive values where no value is coprime to all others, extending known results from linear cases.

## Contribution

It establishes a new coprimality condition for polynomial values, generalizing previous linear polynomial results to quadratic and cubic cases.

## Key findings

- Existence of an integer G_f such that for all k ≥ G_f, infinitely many n satisfy the coprimality condition.
- Extension of coprimality results from linear to quadratic and cubic polynomials.
- Identification of conditions under which polynomial values share common factors across consecutive sequences.

## Abstract

Let $f\in\mathbb{Z}[X]$ be quadratic or cubic polynomial. We prove that there exists an integer $G_f\geq 2$ such that for every integer $k\geq G_f$ one can find infinitely many integers $n\geq 0$ with the property that none of $f(n+1),f(n+2),\dots,f(n+k)$ is coprime to all the others. This extends previous results on linear polynomials and, in particular, on consecutive integers.