[lisper]

[Not the OP but I think I understand it well enough to take a whack at an ELI5.]

Elliptic curves are a particular kind of cubic equation, exactly like the quadratic equations you studied in junior high algebra, except with one term being raised to the third power instead of just squared (and a few other conditions). It turns out that these equations have vastly more complicated behavior than quadratics and give rise to a whole host of problems that mathematicians are still working to solve. One of the interesting problems arises when you ask: what are the solutions to the equation if we restrict ourselves only to rational numbers? It turns out that rational solutions to elliptic curve equations can be grouped into families of solutions where each member of the family can be derived from other members by linear operations (addition and multiplication by a constant). The number of such families of solutions is called the rank of the equation. (Note: it's actually a little more complicated than that, but that's the gist of it. See [1] if you want the details.)

It is observed empirically (by solving lots of elliptic curve equations) that the rank tends to be small. Indeed, the elliptic curve that made the news did so because it has a rank of 29, the largest rank currently known. But no one knows if this is the biggest possible (almost certainly not) or if there is an upper bound on the possible rank of an elliptic curve. Solving that would win you a Fields medal.

(Note: there are results on the upper bound of the average rank of families of elliptic curves [2] but that is not the same as an absolute upper bound.)

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[1]https://en.wikipedia.org/wiki/Rank_of_an_elliptic_curve

[2] https://en.wikipedia.org/wiki/Rank_of_an_elliptic_curve#Uppe...

[jjice]

This is a fantastic ELI5, thank you!

[lisper]

Thanks! I try hard to produce quality technical pedagogy, so you just made my day.

[ElijahLynn]

Not exactly the 5 year old level though, "you studied in junior high algebra".

Thank you for sharing, and I'm still looking for ELI5 though, because I don't remember algebra class that well.

[eddd-ddde]

For the longest time I thought elliptic curves where quadratic curves.

Wouldn't it had been more accurate to name them elliptic surfaces?

[lisper]

The name derives from the fact that they originally arose in connection with trying to determine the arc length of an ellipse. See:

https://people.math.rochester.edu/faculty/doug/mypapers/wayn...

[QuesnayJr]

Just to be clear, an ellipse is a quadratic curve. Ellipses are not elliptic curves. (They are still curves, though, as long as you restrict to plugging in real numbers, not complex.) The terminology is unfortunate.

[CarpaDorada]

They're curves (one-dimensional), not surfaces. An example of an elliptic curve is y^2 = x^3 + 1. The polynomial P(x,y) = x^3 + 1 - y^2 has degree 3. A surface is a 2 dimensional geometric shape.

[fsckboy]

>Solving that would win you a Fields medal

it would not win me a Fields medal: ageism, it's only for under 40s.

[acer4666]

Youd probably get the Abel prize (which has a significantly larger cash prize)

[btreecat]

I strongly doubt that's the primary factor preventing you winning.