262 CHAPTER 23
right triangle has area 6, so 6 is a congruent number. The 5-
12-13 right triangle has area 30, so 30 is a congruent number.
The right triangle with sides
has area 5, so 5 is a con-
gruent number. It can be proven that 5 is the smallest congruent
The congruent number problem is a generalization of the
Diophantine–Pythagorean problem, which we discussed in chap-
ter 9 on elliptic curves. Remember the question: “What are the
possible right triangles of area 1 that have all their sides of rational
length?” So we were then asking, “Is 1 a congruent number?”
Some algebraic manipulations reduced the problem to solving a
cubic equation, which is equivalent to studying an elliptic curve.
Similarly, for the general congruent number problem, we end up
studying certain elliptic curves.
Again, as in the proof of Fermat’s Last Theorem, the repre-
sentations of the absolute Galois group of Q that arise from the
torsion points on these elliptic curves play a crucial role, as does a
certain modular form. Tunnell in (Tunnell, 1983) used these ideas
to give an easily computable property that a congruent number n
has to satisfy. So we can (provably) rule out a given n, if it fails
this criterion. Also, as we mentioned in chapter 20, if a certain
conjecture about elliptic curves, called the Birch–Swinnerton-Dyer
Conjecture, holds, then Tunnell’s criterion is “if and only if” and
could be used to rule in or out any given n. Thus, a proof of the
Birch–Swinnerton-Dyer Conjecture would also completely settle
the congruent number problem.
In both this example and the proof of Fermat’s Last Theorem,
elliptic curves and the Galois representations that they deﬁne
are at the forefront. Partly this is because mathematicians under-
stand the relatively simple two-dimensional representations of the
absolute Galois group of Q that arise in these cases much better
than general, higher-dimensional representations. They have
proved good reciprocity laws with modular forms for some of these
two-dimensional representations, and this has enabled work to
proceed to a much deeper level in such problems, than (so far) for
general systems of polynomial Z-equations.