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RETROSPECT 261
institutions and transactions on the Internet. Number theorists
have been starting cryptography companies and earning lots of
money—perhaps the clearest indication, in the United States at
least, of “reality.”
Still, in the end, we ﬁnd ourselves drawn to the beauty of the pat-
terns themselves, and the amazing fact that we humans are smart
enough to prove even a feeble fraction of all possible theorems
about them. Often, greater than the contemplation of this beauty
for the active mathematician is the excitement of the chase. Trying
to discover ﬁrst what patterns actually do or do not occur, then
ﬁnding the correct statement of a conjecture, and ﬁnally proving
it—these things are exhilarating when accomplished successfully.
Like all risk-takers, mathematicians labor months or years for
these moments of success.
And yet, every encounter with the exotic leads to a lifting of the
veil and the destruction of a mystery. Mathematicians often get
bored by a problem after they have fully understood it and have
given proofs of their conjectures. Sometimes they even forget the
precise details of what they have done after the lapse of years,
having refocused their interest in another area. The common notion
of the mathematician contemplating timeless truths, thinking over
the same proof again and again—Euclid looking on beauty bare—is
rarely true in any static sense.
Luckily, the mathematical world is so rich and complex that we
need not fear running out of patterns to wonder about.
The Congruent Number Problem
For a ﬁnal example of an interesting problem in number theory
that can be attacked using the ideas in this book, we discuss the
“congruent number problem. This problem goes back to ancient
Greek mathematics, but it was only in 1983 that a lot of progress
was made in solving it. It is still not totally solved.
The congruent number problem asks: What integers can be
areas of right triangles, all of whose sides are rational numbers?
We call these areas “congruent numbers. For example, the 3-4-5 262 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
3
2
-
20
3
-
41
6
has area 5, so 5 is a con-
gruent number. It can be proven that 5 is the smallest congruent
number.
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.

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