On Simultaneous Arithmetic Progressions on Elliptic Curves

and we consider two equations related by such a change of variables to represent the same curve (equivalently, we will deal with elliptic curves up to so-called Weierstrass changes of variables). Consider P0, . . . , Pn ∈ E(K), with Pi = (xi, yi) such that x0, . . . , xn is an arithmetic progression. We say that P0, . . . , Pn are in x-arithmetic progression (x-a.p.) and also say that E has an x-arithmetic progression of length n+1. This does not depend on the Weierstrass equation considered. The same definition goes for y-arithmetic progressions (y-a.p.). However, in this case, changes of variables (even those that preserve Weierstrass equations) can create and remove y-arithmetic progressions.