Let $E/\mathbb {Q}$ be an elliptic curve. For a prime p of good reduction, let $r(E,p)$ be the smallest non-negative integer that gives the x-coordinate of a point of maximal order in the group $E(\mathbb {F}_p)$ . We prove unconditionally that $r(E,p)> 0.72\log \log p$ for infinitely many p, and $r(E,p)> 0.36 \log p$ under the assumption of the Generalized Riemann Hypothesis. These can be viewed as elliptic curve analogues of classical lower bounds on the least primitive root of a prime.

中文翻译：

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Pillai 原根下界的椭圆曲线模拟

令 $E/\mathbb {Q}$ 是一条椭圆曲线。对于良好约简的素数p，令 $r(E,p)$ 是给出组 $E(\mathbb {F}_p)$ 中最大阶点的x坐标的最小非负整数. 在广义黎曼假设的假设下，我们无条件地证明 $r(E,p)> 0.72\log \log p$ 对于无限多个p，并且 $r(E,p)> 0.36 \log p$ 。这些可以看作是素数的最小原始根上经典下界的椭圆曲线类似物。