An elliptic curve analogue of Pillai’s lower bound on primitive roots
Canadian mathematical bulletin, Tome 65 (2022) no. 2, pp. 496-505
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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.
Jin, Steven; Washington, Lawrence C. An elliptic curve analogue of Pillai’s lower bound on primitive roots. Canadian mathematical bulletin, Tome 65 (2022) no. 2, pp. 496-505. doi: 10.4153/S0008439521000448
@article{10_4153_S0008439521000448,
author = {Jin, Steven and Washington, Lawrence C.},
title = {An elliptic curve analogue of {Pillai{\textquoteright}s} lower bound on primitive roots},
journal = {Canadian mathematical bulletin},
pages = {496--505},
year = {2022},
volume = {65},
number = {2},
doi = {10.4153/S0008439521000448},
url = {http://geodesic.mathdoc.fr/articles/10.4153/S0008439521000448/}
}
TY - JOUR AU - Jin, Steven AU - Washington, Lawrence C. TI - An elliptic curve analogue of Pillai’s lower bound on primitive roots JO - Canadian mathematical bulletin PY - 2022 SP - 496 EP - 505 VL - 65 IS - 2 UR - http://geodesic.mathdoc.fr/articles/10.4153/S0008439521000448/ DO - 10.4153/S0008439521000448 ID - 10_4153_S0008439521000448 ER -
%0 Journal Article %A Jin, Steven %A Washington, Lawrence C. %T An elliptic curve analogue of Pillai’s lower bound on primitive roots %J Canadian mathematical bulletin %D 2022 %P 496-505 %V 65 %N 2 %U http://geodesic.mathdoc.fr/articles/10.4153/S0008439521000448/ %R 10.4153/S0008439521000448 %F 10_4153_S0008439521000448
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