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No credit card required 254 CHAPTER 22
we are trying to prove in the ﬁrst place, so it would seem unlikely
that we could verify it a priori.
This is where the miracles start happening.
10
By a theorem
of Robert Langlands and Jerry Tunnell (Tunnell, 1981) from the
theory of “automorphic representations, we know, by completely
different methods from anything discussed in this book that if
v = 3, then ψ
E,3
is automatically modular! So if ψ
E,3
is irreducible,
we can apply Theorem 22.6 for v = 3, and so
E,3
is modular.
But what if ψ
E,3
is not irreducible? Wiles came up with a very
clever argument that if ψ
E,3
is not irreducible, there is another
elliptic curve E
for which ψ
E
,3
is irreducible, and for which ψ
E,5
=
ψ
E
,5
. It follows that
E
is modular and hence that ψ
E
,5
is modular
and hence that ψ
E,5
is modular. It also turns out that if ψ
E,3
is
not irreducible, ψ
E,5
must be irreducible! So now we can apply
Theorem 22.6 for v = 5.
Either way, the Frey curve turns out to satisfy the Modularity
Conjecture and FLT is proven true. Perhaps even more signiﬁ-
cant than FLT is the Modularity Conjecture itself, which as we
mentioned has now been proven for all elliptic curves.
Generalized Fermat Equations
Now that FLT has been proven, number theorists are moving on to
other equations. An equation of the form
x
p
+ y
q
= z
r
where x, y, and z are unknown integers, and where the three
exponents p, q, and r are not all the same, is called a generalized
Fermat equation. Because it looks like the equation in FLT, it might
be expected that similar methods could be used to solve it. This is
partially true. But the fact that the three exponents p, q, and r are
not all the same makes it even more difﬁcult than FLT.
To make progress on these generalized Fermat equations, math-
ematicians usually make some assumptions to narrow the scope of
10
We mean “start” from a logical, not historical, point of view; these miraculous
mathematical facts were known before Wiles proved his theorem.

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