Draw a line between P and Q. That line will intersect the curve at a third point, R. (Mathematicians have a special trick for dealing with the case where the line doesn’t intersect the curve by adding a “point at infinity.”) The reflection of R across the x-axis is your sum P + Q. Together with this addition operation, all the solutions to the curve form a mathematical object called a group.
Mathematicians use this to define the “rank” of a curve. The rank of a curve relates to the number of rational solutions it has. Rank 0 curves have a finite number of solutions. Curves with higher rank have infinite numbers of solutions whose relationship to one another using the addition operation is described by the rank.
Ranks are not well understood; mathematicians don’t always have a way of computing them and don’t know how big they can get. (The largest exact rank known for a specific curve is 20.) Similar-looking curves can have completely different ranks.
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