论文标题
小残留类的亚产物
Subproducts of small residue classes
论文作者
论文摘要
对于任何Prime $ p $，令$ y（p）$表示最小的整数$ y $，以便每个减少的残基类$ \ pmod p $由$ \ {1，\ dots，y \} $的某个子集的产品表示。很容易看出，$ y（p）$至少与最小的二次非救助$ \ pmod p $一样大；我们证明$ y（p）\ ll_ \ varepsilon p^{1/（4 \ sqrt e）+\ varepsilon} $，从而增强了伯吉斯的经典结果。该结果是另外两个结果之间的中间力量，即Burthe的证据表明，整数最高$ o_ \ varepsilon（p^{1/（4 \ sqrt e）+\ \ varepsilon} $ ymunsch and shparlinski $ shods $ shods $ shods $ shods $ fallinski $ shods $ shods $ shods $ shods $ shods $ shods $ fallinski的pl od fallinski的p^p^pmod p $ p $ p^{p^{1/（p^{1/（4 \ sqrt e），以及red off fallinski的p。素数的子集最多可达$ o_ \ varepsilon（p^{1/（4 \ sqrt e）+\ varepsilon} $。与后者的结果不同，我们的证明是基本的，与伯吉斯的结构相似，与伯吉斯的证明是最小四边形的非沉利性。
For any prime $p$, let $y(p)$ denote the smallest integer $y$ such that every reduced residue class $\pmod p$ is represented by the product of some subset of $\{1,\dots,y\}$. It is easy to see that $y(p)$ is at least as large as the smallest quadratic nonresidue $\pmod p$; we prove that $y(p) \ll_\varepsilon p^{1/(4 \sqrt e)+\varepsilon}$, thus strengthening Burgess's classical result. This result is of intermediate strength between two other results, namely Burthe's proof that the multiplicative group $\pmod p$ is generated by the integers up to $O_\varepsilon(p^{1/(4 \sqrt e)+\varepsilon}$, and Munsch and Shparlinski's result that every reduced residue class $\pmod p$ is represented by the product of some subset of the primes up to $O_\varepsilon(p^{1/(4 \sqrt e)+\varepsilon}$. Unlike the latter result, our proof is elementary and similar in structure to Burgess's proof for the least quadratic nonresidue.
