Geodesic
Arithmetic progressions on conics
Journal of integer sequences, Tome 20 (2017) no. 2
In this paper, we look at long arithmetic progressions on conics. By an arithmetic progression on a curve, we mean the existence of rational points on the curve whose $x$-coordinates are in arithmetic progression. We revisit arithmetic progressions on the unit circle, constructing 3-term progressions of points in the first quadrant containing an arbitrary rational point on the unit circle. We also provide infinite families of 3-term progressions on the unit hyperbola, as well as conics $ax^{2} + cy^{2} = 1$ containing arithmetic progressions as long as 8 terms.
Classification : 11B25, 11D09
Keywords: arithemetic progression, conic 
@article{JIS_2017__20_2_a6,
     author = {Ciss,  Abdoul Aziz and Moody,  Dustin},
     title = {Arithmetic progressions on conics},
     journal = {Journal of integer sequences},
     year = {2017},
     volume = {20},
     number = {2},
     zbl = {1420.11023},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/JIS_2017__20_2_a6/}
}
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Ciss,  Abdoul Aziz; Moody,  Dustin. Arithmetic progressions on conics. Journal of integer sequences, Tome 20 (2017) no. 2. http://geodesic.mathdoc.fr/item/JIS_2017__20_2_a6/