The Birch and Swinnerton-Dyer Conjecture

Next up is the Birch and Swinnerton-Dyer conjecture – somehow seems an apt name though I’m not really sure why… Anyhow this popped up in Andrew Wiles famous proof of Fermat’s Last Theorem. Fermat was great. He did lots of incredibly difficult and complicated maths in the margins of a textbook without ever showing any of his working out. It took mathematicians almost 400 years to figure it all out and I reckon he is probably the reason that maths teachers today insist that you show your working.

Back to the BSD conjecture (that’s its new hip name). It looks at equations that describe a particular type of graph. For a graph that is just a single straight line we have the general equation y = mx + c, where m is the gradient of the line and c is the intercept with the y-axis. Fancier graphs called elliptic curves also have equations describing them they just happen to be a little more complicated. To get an idea of what an elliptic curve is, just take a pencil (or heaven forbid a pen – drawing graphs with a pen is a big no-no in maths for some reason) and draw a nice squiggly line that has no sharp corners and doesn’t cross over itself. That is your very own elliptic curve (or near enough). Here are some actual elliptic curves for reference.

533px-EllipticCurveCatalog.svg copy

The BSD conjecture (catchy isn’t it?) asks whether or not the equations for these elliptic curves have solutions (the points where they cross the axes) that are whole numbers or fractions. Taking whole numbers and fractions together gives you a group of numbers called rational numbers. Almost every number you can think of is a rational number: 1, 2, 3, ½, ¼, 100 etc. but they are actually less common that the other type of number – irrational numbers. Irrational numbers are everything else that isn’t a whole number or a fraction. So here we mean pi, e, the square root of 2, the golden ratio, there are a few very famous ones and literally an unlimited amount of others.

In maths if we have a fixed number of something we say it is finite: no matter how few or how many we have, we are able to put them in some kind of order and count them up. It might take us forever to count them, say if we wanted to count every living thing on earth, but the idea is that if we kept going and didn’t stop (or die) then eventually we would have counted everything alive on earth and arrived at a final number. If we can’t count everything then we say it is infinite. The most obvious example are numbers themselves, or even just whole positive numbers. If you started counting right now you could continue forever. There is no end point, you would just keep counting higher and higher and higher and higher. You could have started counting at the beginning of the universe and you would still be going today. Numbers go on forever and for that reason they are infinite.

We are almost there so let’s take stock. We have fun squiggly curly graphs called elliptic curves. These are described by equations called elliptic equations. These equations can sometimes have rational solutions (the points where they cross the axes), which are fractions or whole numbers. We can have a finite (number we can count) or infinite (goes on forever – we can always find one more) number of these solutions. The BSD conjecture asks whether or not there is a simple way to tell if an elliptic equation has a finite or infinite number of rational solutions. There you go – who’d have thought it was as easy as that? “Simples” as that silly little Russian meerkat would say…


I’ve written a series of articles on each of the 7 Millennium Problems which can be found here.


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