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Fearless Symmetry by Robert Gross, Avner Ash

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110 CHAPTER 9
exponent. Just one for starters and, boom, you get stuck right away.
Voil `a—elliptic curves.
The Congruent Number Problem
Another interesting Greek problem that turns out to be tied up with
elliptic curves is what might be called a Diophantine–Pythagorean
problem: Find all right triangles whose side lengths are rational
numbers and whose area is the integer D. This is also called the
“congruent number problem. It does not sound so difficult, does it?
But it is difficult, and it has not yet been fully solved.
If you want to do a little algebra, you can see how elliptic curves
come up. We look at the case where D = 1. Using only rational
numbers, we want to solve the system of equations:
1. x
2
+ y
2
= z
2
.
2. xy/2 = 1.
These equations correspond to a right triangle whose sides have
lengths |x| and |y|, with hypotenuse |z|. Divide equation (1) through
by z
2
and set X = x/z, Y = y/z. Then our system is equivalent to
1
. X
2
+ Y
2
= 1.
2
. XY/2 = 1/z
2
.
On page 54 in the subsection on Z-equations, we pointed out
that if we set w = 1 + t
2
, the solutions to (1
) are all of the form
X = (1 t
2
)/w and Y = 2t/w, where t can take the value of any
rational number. (Oh yes, there is also the solution X =−1, Y = 0.)
Plugging this into (2
), multiplying through by w
2
, and cancelling
the 2’s, we get
3. t t
3
= (w/z)
2
.
You may think this looks bad, but remember that we can make z
anything we need it to be to make item (3) hold. But we do have
to be able to take the square root of the left side of (3) in order to
solve for z. In other words, the Diophantine–Pythagorean problem
we started with is equivalent to finding all rational numbers t such

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