It is possible to describe all Pythagorean triples by a formula.
Suppose a=m^{2}n^{2},
b=2mn and c=m^{2}+n^{2},
with m and n positive integers with m > n.
Then
(a,b,c) is a Pythagorean triple, and all Pythagorean
triples arise in this way. The proof of that statement requires two
steps. The easy direction is doing the algebra to show that
(a,b,c) is a Pythagorean triple. The harder direction
is showing that all Pythagorean triple arise in this way.
There are several ways to do this, such as rearranging the
equation to say a^{2}=c^{2}b^{2} and then factoring both sides.
Elliptic curves
The connection with elliptic curves comes when we also require
that the area of the triangle is an integer. But first
we explain what is an elliptic curve.
There are many good introductions to elliptic curves,
such as Silverman and Tate's Rational Points on Elliptic Curves,
or Koblitz' slightly more advanced book (which has an extensive
discussion about congruent numbers) Introduction to Elliptic Curves and Modular Forms.
For our purposes we give
an algebraic description which unfortunately hides the beauty
and depth of the subject.
An elliptic curve is an equation in
two parameters which can be rearranged (possibly after a change of variables)
to be in the form
(1) y ^{2} = x^{3}+A x + B,
where A
and
B
are rational numbers.
Technical note: An equivalent definition is
y^{2}= p(x),
where p(x) is a polynomial of degree 3 or 4.
The change of variables y = y' + α x^{2} shows that
if the polynomial p(x) has degree 4, then it can be replaced by
a polynomial of degree 3. Then another changes of variables
will eliminate the x^{2} term, leaving an
equation of the form (1). It is also required that the curve not be singular,
which in form (1) says that 4A^{3}27B^{2} is not equal to 0.
Details in any book
on elliptic curves.
Equation (1) is easy to understand of we think of x
and y as real numbers: the real solutions
to the equation look like one of the following graphs:
Things get interesting, and it becomes a number theory problem, if we
instead look only for rational solutions to equation (1).
Depending on A
and
B, the equation might have no rational solutions,
or just a few, or infinitely many.
A particular case concerns congruent numbers.
The congruent number curve
Suppose we have a right triangle with sides
a, b, c so that
a^{2}+b^{2}= c^{2},
and its area is n = a b/2. If we let
x = (c/2)^{2}
and
y=(b^{2}  a^{2})c/8,
then a little algebra finds that
y^{2} = x^{3} n^{2} x.
Furthermore (this is a simple but important point),
if a, b, and c are rational numbers,
then so are x and y, and y
is not 0. (If y = 0, then either c = 0
and we don't actually have a triangle, or
b^{2} = a^{2} in which case
a = c /√2,
so either a or c is irrational.)
Thus, if n is a congruent number, then the
elliptic curve y^{2} = x^{3} n^{2} x
has a rational point where y is not 0.
It follows that such a point must have infinite order,
which is why the congruent number problem has a connection
to the Birch and SwinnertonDyer conjecture.
