标题： 元素在多项式增长的正则序列中的互质性 显示英文标题

标题： Coprimality of elements in regular sequences with polynomial growth

摘要： 在某些算术序列中的素数研究是数论中的一个基本问题，特别是寻找不同素数的块近年来引起了广泛关注。 在此背景下，我们证明了在某些正规序列中存在长的$k$-wise 互素元素块。 更准确地说，我们证明了对于任何正整数$H \geq k \geq 2$，以及满足$\lim_{x \to \infty} f^{(k)}(x) = 0$和$\limsup_{x \to \infty} f^{(k-1)}(x) = \infty$的实值$k$次连续可微函数$f \in \mathcal{C}^k\left( [1, \infty)\right)$，存在无限多个正整数$n$，使得对于任何整数$1 \leq i_1 < i_2 < \cdots < i_k \leq H$，$$ \gcd\left( \lfloor f(n+i_1)\rfloor, \lfloor f(n+i_2)\rfloor, \cdots, \lfloor f(n+i_k)\rfloor \right) ~=~ 1 $$成立。 此外，我们证明存在一个子集$\mathcal{A} \subseteq \mathbb{N}$，其上Banach密度为一，使得$$ \gcd\left(\lfloor f(n_1) \rfloor, \lfloor f(n_2) \rfloor, \cdots, \lfloor f(n_k) \rfloor\right) ~=~ 1 $$对于任何不同的整数$n_1, n_2, \cdots, n_k \in \mathcal{A}$。 显示英文摘要

摘要： The investigation of primes in certain arithmetic sequences is one of the fundamental problems in number theory and especially, finding blocks of distinct primes has gained a lot of attention in recent years. In this context, we prove the existence of long blocks of $k$-wise coprime elements in certain regular sequences. More precisely, we prove that for any positive integers $H \geq k \geq 2$ and for a real-valued $k$-times continuously differentiable function $f \in \mathcal{C}^k\left( [1, \infty)\right)$ satisfying $\lim_{x \to \infty} f^{(k)}(x) = 0$ and $\limsup_{x \to \infty} f^{(k-1)}(x) = \infty$, there exist infinitely many positive integers $n$ such that $$ \gcd\left( \lfloor f(n+i_1)\rfloor, \lfloor f(n+i_2)\rfloor, \cdots, \lfloor f(n+i_k)\rfloor \right) ~=~ 1 $$ for any integers $1 \leq i_1 < i_2 < \cdots < i_k \leq H$. Further, we show that there exists a subset $\mathcal{A} \subseteq \mathbb{N}$ having upper Banach density one such that $$ \gcd\left(\lfloor f(n_1) \rfloor, \lfloor f(n_2) \rfloor, \cdots, \lfloor f(n_k) \rfloor\right) ~=~ 1 $$ for any distinct integers $n_1, n_2, \cdots, n_k \in \mathcal{A}$.