FERMAT’S LAST THEOREM AND GENERALIZED FERMAT EQUATIONS 243

some idea of how the proof of Fermat’s Last Theorem uses

the “fearless symmetries” of the Galois group that have

been our theme.

The Three Pieces of the Proof

In the last chapter, we gave our last look at reciprocity laws per se.

Now we will show how these laws can be applied to prove Fermat’s

Last Theorem (referred to as FLT throughout this chapter) and

similar theorems.

Before Andrew Wiles burst upon the scene, there were a number

of proposed strategies for proving FLT. Let us begin by explaining

the strategy that Wiles ultimately made to work. This strategy is

composed of several difﬁcult theorems plus one big conjecture. The

proof was completed by Wiles and Taylor–Wiles by proving enough

of the big conjecture to make the whole strategy work.

2

Before we start, we review the statement of FLT:

If n is an integer greater than 2, there are no solutions to

the equation x

n

+ y

n

= z

n

, where we require x, y, and z to be

nonzero integers.

The ﬁrst step is to note that if we can prove this for the exponents

4 and odd prime numbers, then it is true for any n > 2. Why? Given

any integer n > 2, there are two possibilities:

1. If n is divisible by 4, we can write n = 4m, and then the

equation a

n

+ b

n

= c

n

becomes (a

m

)

4

+ (b

m

)

4

= (c

m

)

4

.Ifwe

have shown that there are no solutions with the exponent

4, then this equation has no solutions.

2. If n is not divisible by 4, then it must be divisible by an odd

prime p. Write n = pm, and then the equation a

n

+ b

n

= c

n

becomes (a

m

)

p

+ (b

m

)

p

= (c

m

)

p

. If we have shown that there

are no solutions with the exponent p, then this equation

has no solutions.

2

Wiles collaborated with Richard Taylor to prove one key step in the whole proof. Thus,

the proof of FLT appeared in two papers, one by Wiles (Wiles, 1995) and a shorter one

that was a joint work of Taylor and Wiles (Taylor and Wiles, 1995).

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