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Activity 2-15: Elliptic curves
www.carom-maths.co.uk
There is a famous story about the mathematicians Ramanujan and Hardy.
One day when Hardy visited, he took a cab to Ramanujan’s home.
Stuck for a little small talk as they greeted each other, Hardy said, ‘The number of that cab was 1729 – rather an
uninteresting number, don’t you think?’
To which Ramanujan replied, ‘Not at all, 1729 is the smallest number that can be expressed as
two cubes in two different ways.’
Look at the spreadsheet below to see that Ramanujan was correct:
Two cubes spreadsheet
Task: what is the next highest number to be the sum of two cubes in two different ways?
1729 = 93 + 103 = 123 + 13
We could see the above as two integer solutions to the equation x3 + y3 = 1729.
This is an example of an elliptic curve.
http://www.s253053503.websitehome.co.uk/carom/carom-files/carom-2-13.xlsx
Any cubic curve that doesn’t do unusual things like cross itself can be called an elliptic curve.
Elliptic curves are exceptionally useful in number theory; more and more applications for them have been discovered in recent years.
Suppose we find the equation of the line AB in the diagram.
Coordinate geometry gives that 19
1210
1
12
x
y
which yields that
They were the central tool for Andrew Wiles as he set about proving Fermat’s Last Theorem.
How many times does this line cut the curve?
Putting y = mx + c into x3 + y3 = 1729 gives the cubic equation x3 + (mx + c)3 = 1729.
All cubic equations have either one or three (counting repeated roots separately) real solutions.
Now we know the cubic has two real solutions (which are these?) so it must have a third.
In our case we have
We know two of the roots (1 and 9) so the equation can be written (x 1)(x 9)(x k) = 0.
So and so
Notice that C must be a point with rational coordinates,(a rational point)
since m and c are rational in y = mx +c.
And of course, we can carry on joining up rational points and finding other rational points on the curve
for as long as we wish.
Something amazing; it makes sense to defining ‘adding’ rational points
on the elliptic curve like this:
Given A and B, find the third point C on the curve that is also on AB.
Now reflect C in the axis of symmetry of the elliptic curve
to get the point – C (which is also on the curve.)
We can say that A + B = -C, or A + B + C = 0.
What happens if we add C to –C? We don’t seem to get a third point on the curve here.
In this case, we say C + – C = O, the point at infinity.
In advanced maths, there is a very important structure called a GROUP.
Given a set of objects, S, and a way of combining them, o,there are four rules need to hold for a group to exist:
1. If a and b are in S, then a o b is in S (CLOSURE).
2. If a in S, then a o e = e o a = a for some e in S (the IDENTITY).
3. If a is in S, then a o a-1 = a-1 o a = e for some a-1 in S (the INVERSE of a).
4. If a, b and c are in S, then a o (b o c) = (a o b) o c (ASSOCIATIVITY).
The rational points on an elliptic curve form a group.
1. If you add two rational points on the curve, you get another rational point. TRUE
2. There is a point on the curve so that if you add it to any rational point, it
leaves it unchanged.
The point at infinity works
here = the IDENTITY
Element.
3. Given a rational point, there is another rational point so that when you add the two
together, you get the identity.
Given a point C, the point –C will always do
here.
4. If you add three rational points on the curve,it matters not how you bracket them,
So A + (B + C) = (A + B ) + C (the Associativity rule).
This is hard to prove, but the Geogebra file below demonstrates it.
Click on the buttons at the bottom of the page to work through the construction.
Don’t forget to drag the points at the end!
Geogebra Associativity File
http://www.s253053503.websitehome.co.uk/carom/carom-files/carom-2-15-assoc.ggb
