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Elliptic Curves and Circle Chords | ||
| Iranian Journal of Science | ||
| مقالات آماده انتشار، پذیرفته شده، انتشار آنلاین از تاریخ 28 بهمن 1394 | ||
| نوع مقاله: Research Note | ||
| شناسه دیجیتال (DOI): 10.22099/ijsts.2016.3589 | ||
| نویسندگان | ||
| Arman Shamsi Zargar* 1؛ Farzali Izadi2 | ||
| 1Azarbaijan Shahid Madani University | ||
| 2Urmia University | ||
| چکیده | ||
| Consider the circle (C_R:x^2+y^2=R^2) where (R) varies in the rationals. We show the existence of infinitely many pairs of rational points on $C_R$ so that the corresponding Euclidean distance is rational, and make a connection between the problem of finding such pairs and finding rational points in the plane at rational distance. Further, providing Euclidian distance of any two not necessarily rational points $u, v$ on (C_R) is rational, we relate it to an elliptic curve defined over (Bbb Q(u,v)). Characterizing the notions, we are led to construct infinite families of rank-one elliptic curves (E_{u, v}) over (Bbb Q) with torsion group (Bbb Z/2Bbb ZtimesBbb Z/8Bbb Z) tying the previous result obtained independently by Atkin-Morain cite{A-M}, Kulesz cite{Kul} and Campbell-Goins cite{C-G}. We also find, assuming the parity conjecture, five rank-three examples of these curves. | ||
| کلیدواژهها | ||
| Circle chord؛ Complete solution؛ Elliptic curve؛ General solution؛ (n)-torsion point؛ Rank؛ Rational distance set؛ Rational point | ||
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