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Arithmetic Progressions on Conics

机译:圆锥曲线的算术级数

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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 ax2 + cy2 = 1 containing arithmetic progressions as long as 8 terms.
机译:在本文中,我们研究了圆锥曲线上的长算术级数。曲线上的算术级数是指曲线上x坐标在算术级数上的有理点的存在。我们回顾单位圆上的算术级数,在单位圆上包含任意有理点的第一象限中构造点的3项级数。我们还提供了单位双曲线上无限的3项级数族,以及圆锥形ax2 + cy2 = 1,其中包含长达8个项的算术级数。

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