Department of Mathematics, Bursa Uludağ University, 16059 Bursa, Turkey
*e-mail:* `gamzesavascelik@gmail.com`

Faculty of Engineering and Natural Sciences, Sabancı University, 34956 Tuzla, Istanbul, Turkey
*e-mail:* `mmsadek@sabanciuniv.edu`

Department of Mathematics, Bursa Uludağ University, 16059 Bursa, Turkey
*e-mail:* `gsoydan@uludag.edu.tr`

**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.

**2010 Mathematics Subject Classification.** 11D25, 11G05, 14G05

**Key words and phrases.** Elliptic curve, Edwards curve, Huff curve, rational sequence, rational point

DOI: 10.3336/gm.54.1.04

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