# Rational sequences on different models of elliptic curves

## TL;DR

This paper investigates the existence of algebraic curves over number fields with rational points having specified x-coordinates, providing infinite families of elliptic curves for certain set sizes, thus extending previous results.

## Contribution

It introduces new constructions of elliptic curves with prescribed rational points, generalizing earlier work on algebraic progressions on curves.

## Key findings

- Infinite families of Edwards and Huff curves with specified rational x-coordinates
- Existence results for sets of size 4, 5, or 6
- Extension of previous progressions on algebraic curves

## Abstract

Given a set $S$ of elements in a number field $k$, we discuss the existence of planar algebraic curves over $k$ which possess rational points whose $x$-coordinates are exactly the elements of $S$. If the size $|S|$ of $S$ is either $4,5$, or $6$, we exhibit infinite families of (twisted) Edwards curves and (general) Huff curves for which the elements of $S$ are realized as the $x$-coordinates of rational points on these curves. This generalizes earlier work on progressions of certain types on some algebraic curves.