AI 中文总结
本文研究与有限域本原元相关的丢番图\(m\)元组，给出\(\mathbb{F}_q\)上\(\mathcal{P}\)-丢番图\(m\)元组数量\(N_m\)的渐近公式，并证明当\(q\)足够大时存在这样的\(m\)元组。
AI 中文摘要
受有限域上丢番图元组近期工作的启发，本文考虑与有限域本原元相关的丢番图元组。设\(\mathbb{F}_q\)是含\(q\)个元素的有限域，\(\mathbb{F}_q^*=\mathbb{F}_q\setminus\{0\}\)是其非零元的乘法循环群。若\(g\in\mathbb{F}_q\)生成\(\mathbb{F}_q^*\)，则称\(g\)为本原元。若\(\{x_1,x_2,\cdots,x_m\}\subseteq\mathbb{F}_q^*\)满足对任意\(1\le i\le j\le m\)，\(x_ix_j + 1\)是本原元，则称其为\(\mathbb{F}_q\)上的\(\mathcal{P}\)-丢番图\(m\)元组。记\(N_m\)为\(\mathbb{F}_q\)上\(\mathcal{P}\)-丢番图\(m\)元组的数量，得到渐近公式\(m!\cdot N_m=\left(\frac{\varphi(q - 1)}{q - 1}\right)^{m(m + 1)/2}q^m+O_{m,r}\left(q^{m-\frac{1}{2}+r}\right)\)，其中\(\varphi\)是欧拉函数，\(r\in(0, 1/2)\)。还证明当\(q\ge \exp(\exp(m(m + 1)))\)时存在\(\mathbb{F}_q\)上的\(\mathcal{P}\)-丢番图\(m\)元组。
英文摘要
Inspired by recent works on Diophantine tuples over finite fields, in this paper we consider Diophantine tuples related to primitive elements of finite fields. Let $\mathbb{F}_q$ be the finite field with $q$ elements and $\mathbb{F}_q^*=\mathbb{F}_q\setminus\{0\}$ be the multiplicative cyclic group of all non-zero elements over $\mathbb{F}_q$. An element $g\in\mathbb{F}_q$ is called primitive if $g$ generates the group $\mathbb{F}_q^*$. A set $\{x_1,x_2,\cdots,x_m\}\subseteq\mathbb{F}_q^*$ of $m$ elements is said to be a $\mathcal{P}$-Diophantine $m$-tuple over $\mathbb{F}_q$ if $x_ix_j+1$ is primitive for any $1\le i\le j\le m$. Let $N_m$ denote the number of $\mathcal{P}$-Diophantine tuples over $\mathbb{F}_q$. Then we obtain the asymptotic formula $$m!\cdot N_m=\left(\frac{φ(q-1)}{q-1}\right)^{m(m+1)/2}q^m+O_{m,r}\left(q^{m-\frac{1}{2}+r}\right),$$ where $φ(\cdot)$ is the Euler totient function and $r\in(0, 1/2)$ is an arbitrary real number. Moreover, we prove that there exists a $\mathcal{P}$-Diophantine $m$-tuple over $\mathbb{F}_q$ whenever $q\ge \exp(\exp(m(m+1)))$.
Comments15 pages
