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Elliptic curves II: they really are interesting!
Two similar but very different cases
In the previous instalment (that I recommend you read first if you haven’t done so already) I tried to convince you that there is a natural line of thought that leads us to considering equations of the form y²=x³+k for some integer k. One can, however, lead a horse to the wet stuff but that does not mean that the water sommelier can make it read the menu.
Once we’ve seen that we’ve been naturally led to this case, we might hope that something similar to our analysis of the smaller cases gives us similarly short and sweet arguments. But as soon as we start trying to do some arithmetic in a few small cases, we struggle to spot much of a pattern. This could simply mean we haven’t tried hard enough but believe me, it’s much more involved than that.
If you’re still not convinced, let’s see the full list of solutions in two very similar-looking small cases that turn out to be very different indeed, namely the equations y²=x³+16 and y²=x³+17.
y²=x³+16
You’ve probably quickly spotted a couple of solutions already, afterall 16 is a perfect square. We claim that the equation y²=x³+16 only has the obvious pair of solutions, namely (x, y) = (0, ±4). Moreover, we will do this with an argument so elementary…
