Consider a set of n points on a plane. A line containing exactly 3 out of the n points is called a 3-rich line. The classical orchard problem asks for a configuration of the n points on the plane that maximizes the number of 3-rich lines. In this note, using the group law in elliptic curves over finite fields, we exhibit several (infinitely many) group models for orchards wherein the number of 3-rich lines agrees with the expected number given by Green-Tao (or, Burr, Grünbaum and Sloane) formula for the maximum number of lines. We also show, using elliptic curves over finite fields, that there exist infinitely many point-line configurations with the number of 3-rich lines exceeding the expected number given by Green-Tao formula by two, and this is the only other optimal possibility besides the case when the number of 3-rich lines agrees with the Green-Tao formula.

考虑平面上的一组n点。在n点中恰好包含3点的线称为3富线。经典果园问题要求n的构型平面上的点可以最大化3条富线的数量。在本说明中，使用有限域上椭圆曲线上的群定律，我们展示了果园的几个（无限多个）群模型，其中富3线的数量与Green-Tao（或Burr，Grünbaum）给出的预期数量一致和Sloane）公式以获取最大行数。我们还使用有限域上的椭圆曲线表明，存在无限多的点-线配置，其中3富线的数量比格林-陶道夫公式给出的预期数量多两倍，这是除此以外唯一的最佳可能性3位数的行数与Green-Tao公式一致的情况。