论文标题
有限套件包含近乎主要的根
Finite sets containing near-primitive roots
论文作者
论文摘要
修复$ a \ in \ mathbb {z} $，$ a \ notin \ {0，\ pm 1 \} $。一个简单的参数表明，对于每个$ε> 0 $，几乎全部（渐近100％）Primes $ p $，$ a $ a $ modulo $ p $的乘法订单超过$ p^{\frac12-ε} $。 $ \ frac12 $替换为任何较大的常数，这是一个开放的问题。我们表明，如果$ a，b $是多重独立的，那么对于几乎所有的primes $ p $，$ a，b，ab，a^2b，ab^2 $之一的订单超过$ p^{\ frac {1} {2} {2}+\ frac {1} {1} {30}}}} $。相同的方法允许每个$ε> 0 $生产，显式有限集$ \ MATHCAL {a} $与属性几乎所有Primes $ p $，$ \ Mathcal {a} $的某些元素的订单超过$ p^{1-ε} $。 Modulo通用整数$ n $而不是Primes $ p $的订单相似的结果。
Fix $a \in \mathbb{Z}$, $a\notin \{0,\pm 1\}$. A simple argument shows that for each $ε> 0$, and almost all (asymptotically 100% of) primes $p$, the multiplicative order of $a$ modulo $p$ exceeds $p^{\frac12-ε}$. It is an open problem to show the same result with $\frac12$ replaced by any larger constant. We show that if $a,b$ are multiplicatively independent, then for almost all primes $p$, one of $a,b,ab, a^2b, ab^2$ has order exceeding $p^{\frac{1}{2}+\frac{1}{30}}$. The same method allows one to produce, for each $ε> 0$, explicit finite sets $\mathcal{A}$ with the property that for almost all primes $p$, some element of $\mathcal{A}$ has order exceeding $p^{1-ε}$. Similar results hold for orders modulo general integers $n$ rather than primes $p$.
