A NOTE ON ARITHMETIC PROGRESSIONS ON ELLIPTIC CURVES

Andrew Bremner (Experiment. Math. 8 (1999), 409{413) has described a tech- nique for producing inflnite families of elliptic curves containing length 7 and length 8 arithmetic progressions. This note describes another way to produce inflnite families of el- liptic curves containing length 7 and length 8 arithmetic progressions. We illustrate how the technique articulated here gives an easy way to produce an elliptic curve containing a length 12 progression and an inflnite family of elliptic curves containing a length 9 progression, with the caveat that these curves are not in Weierstrass form.