### Abstract

Several authors have investigated the problem of finding elliptic curves over Q that contain rational points whose x-coordinates are in arithmetic progression. Tradition- ally, the elliptic curve has been taken in the form of an elliptic cubic or elliptic quartic. Moody studied this question for elliptic curves in Edwards form, and showed that there are infinitely many such curves upon which there exist arithmetic progressions of length 9, namely, with x = 0,±1,±2,±3,±4. He asked whether any such curve will allow an extension to a progression of 11 points. This note shows that such curves do not exist. A certain amount of luck comes into play, in that we need only work over a quadratic extension field of Q.

Original language | English (US) |
---|---|

Journal | Journal of Integer Sequences |

Volume | 16 |

Issue number | 8 |

State | Published - Oct 12 2013 |

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### Keywords

- Arithmetic progression
- Chabauty
- Edwards curve
- Elliptic curve

### ASJC Scopus subject areas

- Discrete Mathematics and Combinatorics

### Cite this

**Arithmetic progressions on Edwards curves.** / Bremner, Andrew.

Research output: Contribution to journal › Article

*Journal of Integer Sequences*, vol. 16, no. 8.

}

TY - JOUR

T1 - Arithmetic progressions on Edwards curves

AU - Bremner, Andrew

PY - 2013/10/12

Y1 - 2013/10/12

N2 - Several authors have investigated the problem of finding elliptic curves over Q that contain rational points whose x-coordinates are in arithmetic progression. Tradition- ally, the elliptic curve has been taken in the form of an elliptic cubic or elliptic quartic. Moody studied this question for elliptic curves in Edwards form, and showed that there are infinitely many such curves upon which there exist arithmetic progressions of length 9, namely, with x = 0,±1,±2,±3,±4. He asked whether any such curve will allow an extension to a progression of 11 points. This note shows that such curves do not exist. A certain amount of luck comes into play, in that we need only work over a quadratic extension field of Q.

AB - Several authors have investigated the problem of finding elliptic curves over Q that contain rational points whose x-coordinates are in arithmetic progression. Tradition- ally, the elliptic curve has been taken in the form of an elliptic cubic or elliptic quartic. Moody studied this question for elliptic curves in Edwards form, and showed that there are infinitely many such curves upon which there exist arithmetic progressions of length 9, namely, with x = 0,±1,±2,±3,±4. He asked whether any such curve will allow an extension to a progression of 11 points. This note shows that such curves do not exist. A certain amount of luck comes into play, in that we need only work over a quadratic extension field of Q.

KW - Arithmetic progression

KW - Chabauty

KW - Edwards curve

KW - Elliptic curve

UR - http://www.scopus.com/inward/record.url?scp=84885973381&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84885973381&partnerID=8YFLogxK

M3 - Article

AN - SCOPUS:84885973381

VL - 16

JO - Journal of Integer Sequences

JF - Journal of Integer Sequences

SN - 1530-7638

IS - 8

ER -