Journal of Integer Sequences, Vol. 16 (2013), Article 13.8.5

Arithmetic Progressions on Edwards Curves

Andrew Bremner
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.

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Received August 6 2013; revised version received September 9 2013. Published in Journal of Integer Sequences, October 12 2013.

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