# A quantitative primitive divisor result for points on elliptic curves

• [1] Department of Mathematics University of Toronto Toronto, Canada Current address: Department of Pure Mathematics University of Waterloo Waterloo, Canada
• Volume: 21, Issue: 3, page 609-634
• ISSN: 1246-7405

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## Abstract

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Let $E/K$ be an elliptic curve defined over a number field, and let $P\in E\left(K\right)$ be a point of infinite order. It is natural to ask how many integers $n\ge 1$ fail to occur as the order of $P$ modulo a prime of $K$. For $K=ℚ$, $E$ a quadratic twist of ${y}^{2}={x}^{3}-x$, and $P\in E\left(ℚ\right)$ as above, we show that there is at most one such $n\ge 3$.

## How to cite

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Ingram, Patrick. "A quantitative primitive divisor result for points on elliptic curves." Journal de Théorie des Nombres de Bordeaux 21.3 (2009): 609-634. <http://eudml.org/doc/10901>.

@article{Ingram2009,
abstract = {Let $E/K$ be an elliptic curve defined over a number field, and let $P\in E(K)$ be a point of infinite order. It is natural to ask how many integers $n\ge 1$ fail to occur as the order of $P$ modulo a prime of $K$. For $K=\mathbb\{Q\}$, $E$ a quadratic twist of $y^2=x^3-x$, and $P\in E(\mathbb\{Q\})$ as above, we show that there is at most one such $n\ge 3$.},
affiliation = {Department of Mathematics University of Toronto Toronto, Canada Current address: Department of Pure Mathematics University of Waterloo Waterloo, Canada},
author = {Ingram, Patrick},
journal = {Journal de Théorie des Nombres de Bordeaux},
keywords = {elliptic curve; point of infinite order; quadratic twist},
language = {eng},
number = {3},
pages = {609-634},
publisher = {Université Bordeaux 1},
title = {A quantitative primitive divisor result for points on elliptic curves},
url = {http://eudml.org/doc/10901},
volume = {21},
year = {2009},
}

TY - JOUR
AU - Ingram, Patrick
TI - A quantitative primitive divisor result for points on elliptic curves
JO - Journal de Théorie des Nombres de Bordeaux
PY - 2009
PB - Université Bordeaux 1
VL - 21
IS - 3
SP - 609
EP - 634
AB - Let $E/K$ be an elliptic curve defined over a number field, and let $P\in E(K)$ be a point of infinite order. It is natural to ask how many integers $n\ge 1$ fail to occur as the order of $P$ modulo a prime of $K$. For $K=\mathbb{Q}$, $E$ a quadratic twist of $y^2=x^3-x$, and $P\in E(\mathbb{Q})$ as above, we show that there is at most one such $n\ge 3$.
LA - eng
KW - elliptic curve; point of infinite order; quadratic twist
UR - http://eudml.org/doc/10901
ER -

## References

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