Arithmetic Progressions on Edwards Curves
School of Mathematical and Statistical Sciences
Arizona State University
Tempe AZ 85287-1804
Several authors have investigated the problem of finding elliptic
curves over Q that contain rational points whose x-coordinates are in
arithmetic progression. Traditionally, 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.
Full version: pdf,
Received August 6 2013;
revised version received September 9 2013.
Published in Journal of Integer Sequences, October 12 2013.
Journal of Integer Sequences home page